## Monthly Archives: December 2011

### Answers to the Wrong Reponses

Rebecca Watson is at it again… terrorizing the Internetz with her whole “being right” thing… When will she learn?

Anyways, yesterday she linked to a Reddit thread in which a 15 year old girl (screen name Lunam) posted a picture of herself holding a copy of Carl Sagan’s The Demon Haunted World. The tactic that many chose to take with her was to then post comment after comment about raping her (mostly via sodomy). Rebecca Watson pointed out (I’m paraphrasing here) that this is bad.

Today Greta Christina posted a similar blog (both about this particular event as well as the larger context of sexism that exists) which begins with a list of “Yes but…” comments which try to trivialize the horrible things that get said against women in situations like this.

Both of those posts say everything much better that I possibly could, but I wanted to take a few minutes and respond individually to the “Yes but…” comments that Greta began her article with. I don’t know if they were meant to be rhetorical or if they were direct quotes taken from idiots on the web (I’m really hoping the former) but either way I’ve definitely seen and heard similar statements by people and would like to clarify (for any confused) why they are stupid.

“Yes, but… not all men are like that. And if you’re going to talk about misogyny, you have to be extra-clear about that.”

Why? If you’re reading this post (or, more likely, Rebecca’s or Greta’s) odds are you’re an atheist. I also think it would be fair to bet that you’ve had your complaints about religion in the past. So do you phrase these as “As nice as most priests are, it sure would be swell if those extra-bad priests would stop raping little kids.” or do you get angry and yell from the rooftops that we need to stop the child-raping motherfuckers from ever touching another kid? For a rationally-minded community, you’d think a little thought could go into the idea that yes, we all know not all men are evil, just as not all women are good. Nobody should have to spell this out every time someone wants to make a point about an offence that’s been committed.

“Yes, but… misogyny doesn’t just happen in (X) community (atheist, black, gay, etc.). In fact, it’s worse in some other communities. So it’s not fair to talk about misogyny when it does happen in (X) community, as if it’s something special that we’re doing wrong.”

I’m going to go back to the pedophile analogy from above because hopefully we can all agree that raping children is wrong, and adding to it the recent Penn State scandal: “Child rape doesn’t just happen at Penn State. In fact, many more children are raped by Catholic priests. So it’s not fair to talk about the Penn State incident as if it’s something special that they were doing wrong.” I’m really hoping that sounds as stupid to you as it does to me. Just because worse things are out there doesn’t mean we should be any less vocal about other injustices that exist.

“Yes, but… (X) community where misogyny happens has some great things about it, too. It’s not fair to paint everyone in it with the same brush.”

Your community has some great things going for it? Awesome! Let’s get rid of the shitty parts (in this case, the stuff that makes women not want to join you) and then we can all be awesome together!

“Yes, but… the woman/ women in question could have done something to avoid the misogyny she got targeted with. She/ they could have stayed anonymous/ concealed her gender/ dressed differently/etc. I’m not saying it’s her fault, but…”

You’re partly right here. To take the example of Lunam from Reddit, she could easily have posted just a picture of the book and (possibly) avoided the deluge of sexual comments that followed. The point is why should she have to? Most any guy could post the exact same photo (with himself instead of her) and presumably wouldn’t have been subjected to the barrage that followed for her. But this girl (and anyone like her) needs to hide her face to be taken seriously? There are already places in the world where women aren’t allowed to show their faces, and I suspect that very few of you would actually want to go and live there. If you want to be anonymous on the Internet, you have that option. But you should also be given the option of being allowed to be yourself and taken seriously by those around you.

“Yes, but… the woman/ women in question didn’t behave absolutely perfectly in all respects. Why aren’t we talking about that?”

For two reasons. Because A) not everybody shares the same interpretation of her actions as you, and B) even if we did, it doesn’t necessarily excuse what was done to her by others. Context matters, but there’s very little context that could excuse telling a 15 year old girl that her blood will form a natural lubricant as you sodomize her.

Because the coverage of an incident has absolutely nothing to do with the incident itself. If you doubt someone’s journalistic integrity then that’s one issue, but when you can see the original story free from any potential bias, it shouldn’t affect the actual facts regarding the incident itself.

“Yes, but… there are worse problems in the world. Starving people in Africa, and so on. Why are you complaining about this?”

If you try to pull this excuse then I hope that everyone in your life resolves to repeat this back to you the next time you mention having a headache or stubbing your toe. Bad things are bad. Worse things are bad. Let’s fix both of them.

“Yes, but… gender expectations hurt men, too. Why aren’t we talking about that?”

Because that’s not typically the topic at hand when this gets brought up. Its a non-sequitur. There are articles about this. In 10 seconds of googling I found these two. I’m sure there are more. Why not try posting there? Failing that, why not start your own blog about how gender expectations hurt everyone?

“Yes, but… people are entitled to freedom of speech. How dare you suggest that speech be censored by requesting that online forums be moderated?”

“Yes, but… calling attention to misogyny just makes it worse. Don’t feed the trolls. You should just ignore it.”

By allowing the trolls to shut me up and change who I am, they’ve won anyways. This isn’t about them. This is about otherwise well-meaning men (and women) who have had these attitudes about women ingrained into them from a very early age, and trying to effect social change by pointing out that their outlooks are outdated and illogical. We’re not about to let a couple of trolls stop us from getting our message out there. Besides, trolls can’t regenerate from fire or acid wounds.

Yes. Yes we do. Anger is a powerful tool that is directly responsible for every successful social movement in history. Here’s a video explaining how it works. Yeah, it’s 48 minutes long. Suck it up.

“Yes, but… what about male circumcision?”

It’s bad. What’s your point? Again, this is typically used as a non-sequitur. There are valid points to be made about how horrible male circumcision is, but they need not be made at the expense or the trivialization of other problems in society. You care so much about this issue? Start doing something about it. Spamming the comment sections for unrelated issues just comes across as lazy.

“Yes, but… Rebecca Watson or some other feminist said something mean or unfair in another conversation weeks/ months/ years ago. Why aren’t we talking about that?”

Wait, people aren’t talking about Rebecca Watson? I’m so confused… Did I wake up in the mirror universe again? That could explain the goatee… But seriously, we’re not talking about it because it was weeks/months/years ago and it isn’t related to the topic at hand. Additionally, feminists are not all one huge group. Just because one of them hurt your feelings however long ago doesn’t mean that it invalidates something you’re reading now. Newsflash: you can disagree with someone about one thing and agree with them about another. That’s part of what skepticism is: objective analysis.

“Yes, but… why is it so terrible to ask a woman for coffee in a hotel elevator at four in the morning?”

Because it makes her uncomfortable, she just spend several hours talking about how it makes her uncomfortable, and if your goal is sincere in actually wanting to get to know her over coffee then you should be doing everything in your power not to be making her uncomfortable.

## 2.3 Enumerable sets

Another useful notion we will be using time and again is that of effective enumerability. As with effective decidability and effective computability, we will first explain what it means to be enumerable in principle, before moving on to the effectively so. Straight from the book:

A set $\Sigma$ is enumerable if its members can – at least in principle – be listed off in some order (a zero-th, first, second) with every member appearing on the list; repetitions are allowed, and the list may be infinite.

What this means is that you can give a (possibly infinite) list that will contain every single member of $\Sigma$ and each member is guaranteed to appear a finite number of entries into this list. The finite case is fairly obvious if $\Sigma=\emptyset$ (where $\emptyset$ denotes the empty set containing no elements) then trivially all elements of $\Sigma$ will appear on any list (say, the empty list containing no entries). If $\Sigma$ is larger, but still finite, we can imagine just going through each of the elements and listing them one by one. For example, $0, 1, 2, 3, 4, 5$ is an enumeration of the finite set $\{0,1,2,3,4,5\}$. Similarly, $0, 1, 5, 3, 4, 3, 0, 1, 2$ also enumerates $\Sigma$, although with redundancies and not in a natural order.

The tricky case is infinite lists. The condition you need to pay special attention to in this instance is that each element of $\Sigma$ must appear a finite number of entries into the list. So, for example, the following lists each enumerate $\Sigma=\mathbb N$ (where $\mathbb N$ denotes the infinite set containing all natural numbers):

1. $0, 1, 2, 3, 4, 5,\ldots$
2. $1, 0, 3, 2, 5, 4,\ldots$
3. $0, 1, 2, 2, 3, 3, 3, 4, 4, 4, 4,\ldots$

Notice that in each case (assuming our patterns hold) we can determine exactly how many entries into the list a given number $n$ will appear. In 1. $n$ is in the $n^\text{th}$ position. In 2. $n$ appears $n+1$ entries down the list if $n$ is even, and $n-1$ entries down if $n$ is odd. In the third example $n$ appears $\Sigma_{i=0}^n i$ entries into the list. Note that to make the math easier, we start our counting at zero: thus, the left-most element listed is the “zero-th”, the next is the first, the next is the second and so on. Now 2. and 3. are contrived examples but the point they make is that each $n$ appears a finite number of entries into the list, and we can tell exactly how far into the list it is. Contrast that with the following non-examples:

1. $0, 2, 4, 6, ... , 1, 3, 5, 7,\ldots$
2. $100, 99, 98, 97,\ldots$
3. $1, 9, 0, 26, 82, 0, 13,\ldots$

In 1., all the odd numbers seem to appear an infinite number of places into the list. This clearly violates precisely what we’re looking at. In 2. there’s still an obvious pattern, but any number greater than 100 doesn’t seem to appear at all. Finally, in 3. there’s no clear pattern to how the numbers are being listed. It is entirely possible that this is the beginning of some valid enumeration, but without more information it’s impossible to tell. So despite the fact that $\Sigma$ is enumerable, none of these three lists are valid ways to do so.

So hopefully that gives you a bit of an intuitive notion of the idea of enumerability. For the more formally-inclined, here is how this is defined mathematically:

The set $\Sigma$ is enumerable iff either $\Sigma$ is empty or else there is a surjective (onto) function $f:\mathbb N\rightarrow\Sigma$ (so that $\Sigma$ is the range of $f$). We say that such a function enumerates $\Sigma$.

The text proves that these two definitions are equivalent, but it’s fairly straightforward, so if you’re having trouble seeing it, I suggest sitting down and working out why these two versions of enumerability come out to the same thing. It should be similarly obvious that any subset of $\mathbb N$ (finite or infinite) is also enumerable. However:

Theorem 2.1 There are infinite sets that are not enumerable.

Proof: Consider the set $\mathbb B$ of infinite binary strings (ie: the set containing strings like $011001011001..."$). Obviously $\mathbb B$ is infinite. Suppose, for the purposes of contradiction (also known as reductio) that some enumerating function $f:\mathbb N\rightarrow \mathbb B$ does exist. Then, for example, $f$ will look something like:

$0\mapsto s_0:\underline{0}110010010\ldots\\1\mapsto s_1:1\underline{1}01001010\ldots\\2\mapsto s_2:10\underline{1}1101100\ldots\\3\mapsto s_3:000\underline{0}000000\ldots\\4\mapsto s_4:1110\underline{1}11000\ldots\\\ldots$

The exact values of $s_i$ aren’t important (as we will see) so this example will abstract to the general case. What we are going to do now is construct a new string, $t$, such that $t$ does not appear in the enumeration generated by $f$. We will do this by generating $t$ character-by-character. To determine the $n^\text{th}$ character in $t$ simple look at the $n^\text{th}$ character of $s_n$ and swap it. Thus, given our example enumeration above, the first 5 characters of $t$ would be $01010\ldots"$ which we get by just this method (for convenience, the $n^\text{th}$ character of each $s_n$ has been underlined). Now all we have to do is notice that $t$ will differ from each of the $s_i$‘s at precisely the $i^{th}$ position. As such, $t$ does not appear in the enumeration generated by $f$. Thus, $f$ is not an enumeration of $\mathbb B$ which contradicts our hypothesis that $\mathbb B$ is enumerable.

QED

This gives us some interesting corollaries depending on how you want to interpret the set $\mathbb B$:

For example, a binary string $b\in\mathbb B$ can be thought of as representing a real binary decimal number $0\leq b\leq 1$ (ie: $0010110111..."$ would represent $0.0010110111...$ and $0000000000..."$ would represent $0$. Thus we know that the real numbers in the interval $[0,1]$ are not enumerable  (and so neither is the set of all real numbers $\mathbb R$).

Another way to think of $\mathbb B$ is that it is the set of sets of natural numbers. To see this, interpret a given string $b=b_0b_1b_2\ldots"$ to be the set $b^\prime=\{n|b_n=1\}$, where a number $n\in b^\prime$ iff $b_n=1$ and $n\not\in b^\prime$ iff $b_n=0$. So for example, if $b=10101000111..."$ then $b^\prime=\{0,2,4,8,9,10,\ldots\}$. Thus, the set of sets of natural numbers (denoted $\mathcal P\mathbb N$) is also not enumerable.

In later chapters we will learn the notion of a characteristic function which is a function $f:\mathbb N\rightarrow\{0,1\}$ which takes a numerical property $P$ and maps $n\mapsto 0$ if $P n$ holds and $n\mapsto 1$ if $\neg Pn$ holds. (This may seem backwards, since $0$ typically denotes $\texttt{False}$ and $1$ denotes $\texttt{True}$, however we will see the reasons for this in due course.) If we consider an element $b=b_0b_1b_2\ldots"\in\mathbb B$ to describe a characteristic function $b^\prime$ by $n\mapsto b_n$, then we can observe that the set of all characteristic functions is similarly non-enumerable.

Next time we will finish up chapter 2 by discussing the limitations of what can be effectively enumerated by a computer program.

# 2 Decidability and enumerability

Here we go over some basic notions that will be crucial later.

## 2.1 Functions

As I imagine anyone reading this is aware (although it’s totally cool if you’re not… that’s why it’s called learning), a function $f:\Delta\rightarrow\Gamma$ is a rule $f$ that takes something from its domain $\Delta$ and turns it into something from its co-domain $\Gamma$. We will be dealing exclusively with total functions, which means that $f$ is defined for every element in $\Delta$. Or, more plainly, we can use anything in $\Delta$ as an argument for $f$ and have it make sense. This is contrasted with the notion of partial functions, which can have elements of the domain that $f$ isn’t designed to handle. We will not be using partial functions at any point in this book (or so it promises).

So, given a function $f:\Delta\rightarrow\Gamma$, some definitions:

The range of a function is the subset of the $\Gamma$ that $f$ can possibly get to from elements of $\Delta$, ie: $\{f(x)|x\in\Delta\}$. In other words, the range is the set of all possible outputs of $f$.

$f$ is surjective iff for every $y\in\Gamma$ there is some $x\in\Delta$ such that $f(x)=y$. Equivalently, $f$ is surjective iff every member of its co-domain is a possible output of $f$ iff its co-domain and its range are identical. This property is also called onto.

$f$ is injective iff for it maps every different element of $\Delta$ to a different element of $\Gamma$. Equivalently, $f$ is injective iff $x\neq y$ implies that $f(x)\neq f(y)$. This property is also called one-to-one because it matches everything with exactly one corresponding value.

$f$ is bijective iff it is both surjective and injective. Because $f$ is defined for every element of $\Delta$ (total), can reach every member of $\Gamma$ (surjective) and matches each thing to exactly one other thing (injective), an immediate corollary of this is that $\Delta$ and $\Gamma$ have the same number of elements. This is an important result that we will use quite often when discussing enumerability.

## 2.2 Effective decidability, effective computability

Deciding is the idea of determining whether a property or a relation applies in a particular case. For example, if I ask you to evaluate the predicate “is red” against the term “Mars”, you would say yes. If I gave you the predicate “halts in a finite number of steps” to the computer program $\texttt{while (true);}$ you would probably say no. In either case you have just decided that predicate.

Computing is the idea of applying a function to an argument and figuring out the result is. If I give you the function $f(x)=x+1$ and the argument $x=3$ you would compute the value $4$. If I give you the function $f(x)=\text{the number of steps a computer program }x\text{ executes before halting}$ to the argument of the same computer program as above, you would conclude that the result is infinite. In both cases you have just computed that function.

What effectiveness comes down to is the notion of whether something can be done by a computer. Effective decidability is the condition that a property or relation can be decided by a computer in a finite number of operations. Effective computability is the condition that the result of a function applied to an argument can be calculated by a computer in a finite number of operations. For each notion, consider the two sets of two examples above. In each, the first is effectively decidable/computable and the second is not, for reasons I hope will eventually be clear.

This raises an obvious questions: what is a computer? Or, more to the point, what can computers do exactly? For our purposes we will be using a generalized notion of computation called a Turing machine (named for their inventor, Alan Turing). Despite its name, a Turing machine is not actually a mechanical device, but rather a hypothetical one. Imagine you have an infinite strip of tape, extending forever in both directions. This tape is divided up into squares, each square containing either a zero or a one. Imagine also that you can walk up and down and look at the square you’re standing next to. You have four options at this point (and can decide which to do take based on whether you’re looking at a zero or a one, as well as a condition called the “state” of the machine): you can either move to the square to your left, move to the square on your right, change the square you’re looking at to a zero, or change it to a one. It may surprise you, but the Turing machine I have just described is basically a computer, and can execute any algorithm that can be run on today’s state-of-the-art machines.

In fact, throughout the history of computability theory, whenever a new model has been developed of what could be done algorithmically by a computer (such as $\lambda$-calculus, $\mu$-calculus, and even modern programming languages) it has turned out that each of these notions were equivalent to a Turing machine, as well as each other. Thus, Alan Turing and Alonzo Church separately came up with what is now called the Church-Turing thesis (although the book only deals with Turing, hence “Turing’s thesis”):

Turing thesis: the numerical functions that are effectively computable in an informal sense (ie: where the answer can be arrived at by a step-by-step application of discrete, specific numerical operations, or “algorithmically”)  are just those functions which are computable by a properly programmed Turing machine. Similarly, the effectively decidable properties and relations are just the numerical properties and relations which are decidable by a suitable Turing machine.

Of course we are unable to rigorously define an “intuitive” or “informal” notion of what could be computed, so Turing’s thesis could never be formally proven, however all attempts to disprove it have been thoroughly rebuked.

You might wonder, however, about just how long it might take such a simple machine to be able to solve complex problems. And you would be right to do so: Turing machines are notoriously hard to program, and take an enormous number of steps in order to solve most interesting problems. If we were to actually use such a Turing machine to try and get a useful answer to a question (as opposed to, say, writing a C++ program) it could very realistically take lifetimes to calculate. By what right, then, do we call this “effective”? Another objection to be raised might have to do with the idea of an infinite storage medium, which violates basic engineering principles of modern computer architectures.

Both of these objections can be answered at once: when we discuss computability, we are not so much interested in how practical it is to run a particular program. What interests us is to know what is computable in principle, rather than in practice. The reason for this is simple: when we discover a particular problem that cannot be solved by a Turing machine in a finite number of steps, this result is all the more surprising for the liberal attitude we’ve taken towards just how long we will let our programs run, or how much space we will allow them to take up.

One final note in this section. The way we’ve defined Turing machines, they operate on zeroes and ones. This of course reflects how our modern computers represent numbers (and hence why the Turing thesis refers to “numerical” functions, properties and relations). So how then can we effectively compute functions or decide properties of other things, such as truth values or sentences? This is simple. We basically encode such things into numbers and then perform numerical operations upon them.

For example, most programming languages encode the values of $\texttt{True}$ and $\texttt{False}$ as $1$ and $0$, respectively. We can do the same thing with strings. An immediate example is ASCII encoding of textual characters to numeric values which is standardized across virtually all computer architectures. Later in the book we will learn another, more mathematically rigorous way to do the same.

The rest of this chapter is about the enumerability and effective enumerability of sets, but I’m going to hold off on talking about those until next time.

### Gödel! #1 An Introduction to Gödel’s Theorems 1.0

I am going to rip off something Zach Weiner has been doing on his blog where he’s blogging his way through a few different textbooks. This sounds like an awesome way to get a better understanding out of stuff, so I am going to completely steal the idea from him, including the way he formats his titles (while giving him full credit as my inspiration) and blog my way through some of the textbooks I bought in university but never really actually bothered to crack open. Perhaps more fun for me than for you, but we’ll see how it goes.

The textbook I’m going to start off with is called An Introduction to Gödel’s Theorems written by Peter Smith. From the bits of it I’ve actually made use of, it’s a fairly detailed logic text while still being relatively accessible to anyone with a bit of background. Some of the concepts it seems to take for granted are elementary set theory, introductory logic, and basic computability theory. But most everything we’ll need seems to be covered in the text.

# 1 What Gödel’s Theorems say

## 1.1 Basic arithmetic

A lot of what is going to be covered in the text has to do with basic arithmetic, which is to say the natural numbers (0, 1, 2, etc…) and operations on them (addition, multiplication, etc…). Although this will all be flushed out formally in a few chapters, the natural numbers have a specific starting point, 0, each one has a unique successor, and every number falls into this sequence. But this will all be made formal in short order.

Our bigger concern is the notion of a formalized mathematics. In 1920 mathematician David Hilbert put forward a program to axiomatize all of mathematics into a set of finite, simple and non-controversial mathematical statements. The goal was to lift mathematics up by its bootstraps and prove the completeness and consistency of mathematics from these axioms in order to leave zero doubt as to their correctness.

The idea would be that mathematics could be axiomatized into a theory $T$ (a theory is simply a collection of axioms) that would be (negation) complete, which is to say that for any sentence $\phi$, either $\phi$ or $\neg\phi$ would be provable in $T$.

Thus, in our case, we are considering how one could build a complete theory of basic arithmetic where we could prove (or disprove) conclusively the truth of any claim that could be expressed arithmetically. This is where Gödel’s Theorems come into play…

## 1.2 Incompleteness

… by basically shitting all over the idea. What mathematician Kurt Gödel was able to do in a 1931 paper was present a way to, given a theory $T$ which was sufficiently strong enough to express arithmetic, construct a sentence $\textbf G_T$ such that neither $\textbf G_T$ nor $\neg\textbf G_T$ can be derived in $T$, yet we can show that if $T$ is consistent then $\textbf G_T$ will be true.

Thus, basic arithmetic in its most striped-down form fails to be negation complete, which puts quite a dampener on Hilbert’s program.

The specifics of how $\textbf G_T$ is actually constructed is the subject of most of the text, but the gist of it is this: $\textbf G_T$ encodes the sentence “$\textbf G_T$ is unprovable in $T$“. Thus, $\textbf G_T$ is true iff $T$ can’t prove it. Suppose then that $T$ is sound (ie: cannot prove a false sentence). Then if it were to prove $\textbf G_T$ it would prove a falsehood, which violates soundness. Thus $T$ does not prove $\textbf G_T$ and so $\textbf G_T$ is true. Thus, $\neg\textbf G_T$ is false, which means that $T$ can’t prove it either. Again, how $\textbf G_T$ is constructed is what we’ll be getting to, but this is the gist of Gödel’s First Theorem.

It should also be noted that there isn’t only one such sentence that renders $T$ incomplete. Suppose we decide to augment $T$ by adding $\textbf G_T$ to it, to create a new theory $U=T+\textbf G_T$. We will then be able to construct a new Gödel sentence, $\textbf G_U$ which will be true but unprovable in $U$. Since $U$ encompasses $T$, $\textbf G_U$ will also be unprovable in $T$ and we get a construction of an infinite number of unprovable-in-$T$ sentences.

Thus, arithmetic is not only incomplete, but indeed incompletable.

## 1.3 More incompleteness

This incompletability does not just affect arithmetic. In fact it will also affect any systems which could be used to represent arithmetic. For example, set theory can define the empty set $\emptyset$. Then, form the set $\{\emptyset\}$ containing the empty set, followed by the set containing both these sets $\{\emptyset,\{\emptyset\}\}$ and so on. We get a sequence of the form:

$\emptyset,\{\emptyset\},\{\emptyset,\{\emptyset\}\},\{\emptyset,\{\emptyset\},\{\emptyset,\{\emptyset\}\}\}$

where we can define 0 as $\emptyset$, 1 as the set containing 0, 2 as the set containing 0 and 1, and so on. The successor of $n$ is $n$ unioned with itself (ie: $n\cup\{n\}$), addition is defined as iterated succession, multiplication as iterated addition and all of a sudden you have a theory of arithmetic encompassed in set theory. By Gödel’s First Theorem, then, set theory is also incomplete.

## 1.4 Some implications?

This section deals with some philosophical implications of the First Theorem, but doesn’t delve into enough detail to be worth talking about.

## 1.5 The unprovability of consistency

Any worthwhile arithmetical theory will be able to prove the proposition $0\neq1$. Thus, any decent theory that proves $0=1$ will be inconsistent. Furthermore, an inconsistent theory can prove any proposition, including $0=1$, thus a theory of arithmetic $T$ is inconsistent iff $T$ proves $0=1$. Now, we’ve already established that we can encode facts about the provability of propositions in $T$, thus we have a way to encode the idea that $T$ can’t prove $0=1$, which is to say that $T$ can express its own consistency. We’ll call the sentence that expresses this $\textbf{Con}_T$

From above, we’ve already seen that a consistent theory $T$ can’t prove $\textbf G_T$. Since $\textbf G_T$ is itself the sentence that expresses its own unprovability, we can then express (in $T$) that if $T$ is consistent then $\textbf G_T$ is unprovable by $\textbf{Con}_T\rightarrow\textbf G_T$.

Make sense?

According to the text, it turns out (although we’ve yet to see how) that this sentence $\textbf{Con}_T\rightarrow\textbf G_T$ turns out to be provable within theories with conditions only slightly stronger than those required for the First Theorem. However $\textbf G_T$ must still be unprovable within such theories, and so $\textbf{Con}_T$ must also be unprovable otherwise we would be able to get $\textbf G_T$ by simple modus ponens.

As such, we have Gödel’s Second Incompleteness Theorem: that nice theories that can express a sufficient amount of arithmetic can’t prove their own consistency.

## 1.6 More implications?

The key point that I took from this section is that since we’ve shown that a theory of arithmetic $T$ can’t prove its own completeness or consistency, then it certainly can’t prove the same for a richer theory $T^+$. This rudely defeats what remains of Hilbert’s Programme, as arithmetic isn’t capable of validating itself, let alone the rest of mathematics.

## 1.7 What’s next?

Obviously this is all pretty roughshod. In chapter 2 we go over a few basic notions that we will need to prove things in more detail. Chapter 3 discusses what is meant by an “axiomatized theory”. Chapter 4 introduces concepts specific to axiomatized theories of arithmetic. Chapters 5 and 6 give us some more direction as we head off towards formally proving… pause for dramatic music… Gödel’s First Incompleteness Theorem!

### Godless Bitches – Transgender Episode

This is a reponse to episode 1.13 of the Godless Bitches podcast, whom I absolutely love listening to because it’s all about issues that I don’t normally think about and I love being challenged with new things that I haven’t heard before. Really can’t say enough great things about them.

So first off I want to admit that I know very little about transgender issues (“transgender” or “transgendered”?) but I’m trying to learn, so if you disagree with anything I’m about to say, please let me know because I’m here to learn.

This is primarily a response to a part in the podcast where Beth asks how useful the labels of “female” and “male” or “man” and “woman” are, given the diversity that can exist among the spectrum (around the 29 minute mark). Jen identified them as useful rules of thumb and that we shouldn’t consider them as terms that are necessarily prescriptive of behaviors or even physical characteristics. Natalie agreed, but then went a bit into the point that I thought of when I heard Beth’s question: that “male” and “female” can be useful terms for people to apply to themselves. That’s basically what I want to talk about.

“Man” and “woman” may not have rigid definitions, but I hardly think this disqualifies a word from being useful. I want to draw an analogy. Consider the word “dog”. What does that mean? Look at this picture:

What do those two animals have in common? Very little: they’re different colours, sizes, shapes… But we’re still able to recognize them both as falling under the label “dog” even though we might not have a strict definition of what that term actually means. Yes, I’m sure there’s some strict biological definition of what constitutes a dog, but my point is that even people who lack this education in biology can identify both as dogs. Similarly we can often look at people and recognize them as male or female, even without strict definitions of what these things mean.

Tangent: Does being male mean having a penis? I don’t believe so. Does having a penis make you male? I don’t believe that either. And yet, I consider myself male because I do have a penis. So the attributes we associate with a particular gender are neither necessary nor necessarily sufficient to belong to that gender, but can they be sufficient in certain cases such as my own? For the logic nerds (or possibly just for my own masturbatory needs): $\diamond(\text{penis}\rightarrow\text{male})$ holds, but $\Box(\text{penis}\rightarrow\text{male})$ does not. Thus, $\text{penis}\rightarrow\text{male}$ may hold sometimes, but not always. Does that make sense to anyone else?

Now of course these terms aren’t exclusive and one of the things I’ve learned from reading about transgenderism (is that the right word? spell check says no…) is that gender identity can fall on a spectrum. As I see it, there are two possibilities: multiple people can fall into the same spot on the spectrum, or they can’t. If they can’t then that means that no two people have the same gender identity. This, to me, is absurd and means we can’t actually use labels for gender identities at all, which seems pretty inconvenient to me. Thus, I currently accept the other option, that many people can occupy the same spot on the spectrum. Thus, we can have many people with the gender identity of “male”, many with “female” and many in-betweens. I believe this is where the problem seems to crop up with labels: what words do you use to identify these people? Part of the issue is one of linguistic usefulness. How many people have to have a particular gender identity before we need to come up with a word for it. I don’t think it would be controversial to say that “male” and “female” would be the largest categories. But what then? Do you name a spot on the spectrum that only holds one person? How big does a group need to be before we give it a name? I don’t have the answers, but I hope it at least makes sense as to why I think we need to keep the labels we already have.

This brings me to a point I would like to make on how I feel about gender roles. I’m going to take what I think might be an unpopular stance: Gender roles in society are important. Adhering to them, however, is not. So that we all have our definitions straight (or in case I don’t) when I say “gender roles” I mean any behaviour that is typically associated with either men or women but not both (for example, wearing a suit is more masculine and wearing a dress is more feminine). I think what’s useful about these is that when you want to identify as a particular gender (whether it’s your birth sex or not) these give you patterns you can fall into that will allow other people to more easily identify you as your gender expression (again, am I using that term correctly?). It’s convenient. If I want to be identified as a man, I can grow a beard. If I want to be identified as a woman, I can wear makeup. These gender roles can be extremely useful, for so many reasons! How much of Monty Python’s humour was based on men wearing dresses? What would drag queens wear if there were no such thing as gender-specific clothing?

Where the problem comes in is the expectation that everyone ought to adhere to their gender roles. They may be important, but not important enough to be forced upon people. Girls can like sports, and boys can like dolls and why should it matter to anyone else? It’s not the roles themselves, but rather forcing them upon people unwillingly that becomes harmful. I have a saying I like to use which is that “Girls can be boys too” (or vice-versa, depending on context), which is to say that girls can do things that other people might associate with boys (or the other way around) but that it really shouldn’t matter at all if they do.

There was one other point I wanted to raise, and it’s in reference to a quote I remember from the podcast that I can’t find the time code for, so if I’m misremembering it, I apologize. But someone (I want to say Beth) mentioned not understanding why people have difficulty grasping that biological sex is different from gender identity. I just want to put forward my hypothesis for why I think this is. For thousands of years, people have been treating gender as a binary, biological thing: having a penis makes you a man and having a vagina makes you a woman. Hell, this is even how I learned it as a kid. I think it can be difficult to get past thousands of years worth of linguistic programming to accept that your junk doesn’t determine your gender. Even I still have some problems with this. When I hear the term “transgender male” I think “person who was born with a penis” and I have to stop and thing “wait, no… that’s a person who was born female and now identifies as male”. It’s a problem that I’m slowly getting better at avoiding, but the point is it hasn’t yet become a natural thought process for me. So I’d just like to put out there that maybe some people who have trouble grasping this concept should be cut a little bit of slack. Unless they’re being dicks. Then feel free to verbally lambaste them all across the interwebs.

That’s all I really have to say on the issue. Once again, please feel free to tell me where I’m wrong (or even where I’m right) because this is just how I see it at the moment, and all of it is open to revision as I learn more and more.

### Why I Am an Atheist

PZ Meyers has been doing a cool thing lately on Pharyngula (in addition to the other cool stuff he’s does) posting submitted essays on what brought their writers to atheism. I wanted to share my own.

I was conceived through IVF by two lesbian women, my parents. My donor was anonymous and to this day I have no idea who my biological “father” would be. I don’t think being gay has ever been easy, but today society has made huge strides towards tolerance and acceptance of homosexuality. I was born in 1987 and spent the first ten years of my life experiencing the kind of bigotry that existed back then. When seeking a donor for my conception, my biological mother was told by her doctor that if she wanted to get pregnant she ought to stand on a street corner and get inseminated that way. When I was born, I was not permitted a hyphenated surname from both my mothers, as hyphenate names were only allowed between the mother and father. My biological mother had to change her last name to a hyphenate, have my last name changed as well, and then switch hers back in order to accomplish this. When I was 12, my moms won a 6 year court battle with the government of Alberta to allow my non-biological mother to be legally named as my adoptive mother (from a legal perspective she had previously been my guardian). I am to this day immensely proud of them and the fact that they were able to set this precedent for other gay and lesbian parents to legally adopt children together.

But the point is that I was born with a front-row seat to the kind of persecution that gays and lesbians were experiencing, primarily at the hands of religious right-wing lobby groups.

Whether this actually affected my views of religion is hard to say. I know that my biological mother considers herself a Unitarian, and that my other mom tends to avoid labeling herself when asked. Religion doesn’t come up very often with us. I do remember being dragged to church every couple weeks and sitting there. The only thing I can recall about the experience is an immense sense of boredom and hunger. It finally got to a point where I so desperately did not want to go one week that as my mom was insisting we head out the door that I finally just yelled at her “I DON’T EVEN BELIEVE IN GOD.” This was almost 15 years ago and so I don’t remember the details too clearly, but I don’t think I was made to attend church that day, nor any day after. And for that I am intensely grateful to my parents for not forcing a religion upon me.

But even before that incident, I honestly cannot recall a moment when I believed in God. I would hear stories from adults about how you were supposed to just feel his presence (yes, lowercase ‘h’). How you could tell when he was listening. How he would speak to you if you prayed to him. I felt absolutely none of that.

What you have to understand is the notion of being six or eight years old, having all the adults in your world tell you about this thing that you should feel, and not feeling it at all. I was so confused at what I must have been doing wrong that I couldn’t feel this miraculous presence. Imagine being that young and having every grown-up tell you something that you know to be wrong. It’s extremely alienating. I began to wonder if something was wrong with me. Wanting nobody to know about my inability to sense God, I kept my disbelief a secret until that one day when I had just had enough of sitting in the pews with my stomach growling, having no idea what I was supposed to be listening for in the stories that the boring man at the front of the room was telling.

But even after coming out to my parents and experiencing no wrath or disappointment, I was still alone; confused at how I was the only one who didn’t feel this mystical connection.

Then, when I was roughly 13, my parents bought a VHS copy of a movie they had seen in theaters a couple years earlier. That movie was Dogma, by Kevin Smith. It’s a good movie, but there was one moment in that movie that was an enormous revelation for me. In one of the earlier scenes of the film, Ben Affleck’s uses the word “atheist” (or, as I thought it was said at the time “apheist”), and it was obvious from the context what it meant: a person who doesn’t believe in God.

It was such a small thing, but in that moment it was like being hit with a wave: “There’s a word for it?” I thought to myself, “An actual word that means that I don’t believe in a god? That’s me! And the fact that there’s a word for it? That means I’m not the only one!” Years of loneliness poured off of me as I realized that somewhere in the world there were other people like me who didn’t believe. Not only that, but there were enough of us that English had a word for us. I didn’t know who these other non-believers were, but all of a sudden I knew they were out there, and I felt a connection to them: stronger than the supposed connection with God I was supposed to have felt growing up.

That was the day I was no longer just a kid who didn’t believe: I was an Atheist!

(well… at least I was an Apheist: it wasn’t until early in high school until I actually met a fellow non-believer and learned the right word…)