The formal language of first-order logic allows you to say “for any number” but does not allow you to say “for any set”.
In general, your axioms have to be about numbers, not about sets of numbers.

Sorry, but I am a bit confused: how is this different from having just one axiom that says: “for any set of numbers which fits pattern X, there is a smallest number”? (I am not sure what the pattern is: there are too few examples for a guess.)

More generally, if there is an algorithm that can be used to validate an axiom, then is this algorithm not the “real” axiom, with the axioms being demoted to the level of theorems?

Apologies if this is all in the book: even if I ordered it, it would take weeks to get it. But I’ll take “it’s all in the book” as an answer.

I jumped ahead in the book and read quite a bit more.

I think I’ve pinpointed specifically where we differ. My belief that reality is not able to be described by countably much data stems from my not believing that Determinism is true. Given Determinism is true, I take your point.

Also, I guess since I believe in free will (as you do), but reject determinism (quantum physics and Schrödinger’s cat make me doubt it) I’d be considered a Philosophical Libertarian (nicely rounding out my Political Libertarian side :) )

I’ll have to read up on the Compatibilist philosophers of the day to get greater insight.

On the determinism note, I recommend you checking out “Quantum Enigma” by Bruce Rosenblum and Fred Kuttner http://quantumenigma.com/. They provide some intriguing insights to how physics meets consciousness.

Thanks again for responding to your readers.

In your point 2), you are right that the set of true statements about arithmetic is enumerable. But it is not *recursively* enumerable. In other words, there exists an enumeration of those true statements, but there is no procedure to produce that enumeration. Your formulation fails to notice this distinction.

In your point 4), you are assuming that because “reality” is uncountable it can’t be described by countably much data. But that is false. The real numbers are uncountable, but all true first-order statements about the real numbers can be derived from a countable set of axioms. By contrast, the natural numbers are countable, but no countable set of axioms suffices to imply all true statements about them.

I may have been using my terms improperly.

Specifically, would you agree or disagree with each of these statements:

1. The complexity of a system can be measured by the size of the “data” required to fully describe it.

2. The entirety of what we call mathematics is an infinite enumerable set of truths (or axioms). So the size of the data (complexity) required to describe mathematics is equivalent to the size of data in a first degree infinite set.

3. The size of the data (complexity) of reality is larger than the data in a first degree infinite set. (Time adds a continuum of truths that are innumerable).

4. Therefore reality is more complex than mathematics.

There are many true statements about arithmetic. You are allowed to pick as many of those as you want (even infinitely many) and call them axioms. There’s just one rule, though: You have to give an explicit procedure that will let me distinguish an axiom from a non-axiom in a finite amount of time. One way to do that is to restrict yourself to finitely many axioms. That is not what we usually do, though. Instead, we have infinitely many axioms, but they all fit a certain pattern.

Here’s an axiom: If there are any numbers equal to their own cubes, then there is a smallest one. Here’s another: If there are any even prime numbers, then there’s a smallest one. Here’s another: If there are any numbers not equal to themselves, then there’s a smallest one. Et cetera. Any statement fitting that pattern is an axiom, and this set of axioms obeys the rules because it’s easy to check whether a given statement fits the pattern or not.

Now if you don’t like that set of axioms, you’re welcome to pick another. The only rules are: Your axioms must all be true, and they must be recognizable. That means you’re not allowed to say “I take all true statements as my axioms”. That’ s not a legitimate set of axioms, because I have no finite procedure for recognizing whether a given statement is true or not.

Now Godel’s theorem (in part) says this: You pick your favorite set of axioms. I’ll always be able to find a true statement that does not follow from them.

And thank you Steven for paying attention to our comments. I have not yet re-checked your derivations from the other day, but I will.

Would you say that the “incomplete” set of mathematical axioms (from Godel) is of the first degree of infinity? It would seem so to me.

If that first degree infinite set of axioms represents all of mathematics, then the set of truths that “couldn’t have been different” (and thus couldn’t have been designed) is of the first degree infinity.

In the physical world, there are sets of truths of higher degrees of infinity. Therefore the physical world is more complex than mathematics could ever be.

I guess for me to change my mind I’d have to understand that either the set of mathematical axioms is “larger” than first degree infinite set… or that there doesn’t really exist anything in reality more complex (“larger”) than a first degree infinite set.

Starting with the standard axioms of arithmetic, and armed with the full power of logic, there remain true statements about arithmetic that cannot be proven.

[a] Those true statements cannot be proven by logic plus the standard axioms of arithmetic, but they _can_ be proven, otherwise we would not know that they are true.

[b] One such “true” statement is the statement “this statement cannot be proven by standard arithmetic”. But this is true if, and only if, arithmetic is self-consistent; so we move on to

[c] another such “true” statement: “arithmetic is self-consistent”. To prove this, we need another, more powerful axiomatic system X (and logic). But we cannot prove that X is self-consistent, and therefore the proof is not conclusive.

From [a], [b], and [c] I conclude that truth is, for most practical purposes, the same as provability (although truth within an axiomatic system is not always the same as provability within said system). But only for practical purposes: I believe that the self-consistency of an axiomatic system is an objective fact, but a fact that we cannot know.