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Comment by 12elephant | Hacker News Reader

Your castle is built on sand. What, then, is {}?

One of the axioms of set theory is the axiom of the empty set, which states that there exists at least one set which has no elements. Another axiom of set theory is the axiom of extensionality, which states that two sets are equal if they have the exact same elements: from which it follows that all sets without elements are identical, i.e., there is only one set without elements. We call that the emptyset.

Other axioms of set theory are used to formalize other steps in my post. https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_t... For example, the Axiom of Infinity is used to formalize the handwavy part where I said "In general, the natural n+1 is {0,...,n}".

An axiom is simply something made up.

At the end of the day, all of our Mathematics rests on foundations that are made up.

It's difficult to say what the empty set is. Because it isn't really anything at all.

There are legitimate criticisms you can levy against set theory, but I'm starting to lose you here. I'm not really following your point anymore - this seems like arguing about whether or not mathematics is invented or discovered.

Are you trying to argue coming up with new definitions isn't worthwhile if the axioms don't have a foundation in reality? If so, why? If not, what are you saying?

The original post is called "What even is a number?". The comment I replied to tried to answer that question with "some formalism".

What I'm saying is: questioning foundations leads you to new foundations. New foundations that you can then question all over again. It's turtles all the way down.

If you're actually looking to use math, this is a futile exercise. It works, so just use it. Essentially, I am re-iterating von Neumann's statement:

> In mathematics you don't understand things. You just get used to them.

In that case I'd agree with you. I'm not particularly keen on rehashing foundations of mathematics either.

Do you complain similarly when Euclid states that for any two distinct points, there exists an infinitely long straight line which passes through those two points? We cannot point to that line, we can only point to a small portion of it within our field of vision, and have to extrapolate it to infinity.

Either there is an empty set, or there isn't. If there is one, I win. If not, then let S be the set of all empty sets. There are none, so S is the empty set and I win again.

This is the same response I gave to throwawaymath:

The original post is called "What even is a number?". The comment I replied to tried to answer that question with "some formalism".

What I'm saying is: questioning foundations leads you to new foundations. New foundations that you can then question all over again. It's turtles all the way down.

If you're actually looking to use math, this is a futile exercise. It works, so just use it. Essentially, I am re-iterating von Neumann's statement:

> In mathematics you don't understand things. You just get used to them.

>In mathematics you don't understand things. You just get used to them.

Yep, one of my favorite math quotes and highly under-rated.

One thing I've found is that if you find yourself using the wrong "types", you're almost certainly going in the wrong direction. For example, if you're trying to solve a complex analysis exercise and you find yourself thinking about how the complex numbers = the real-coefficient polynomials modulo the ideal (x^2+1), STOP: You're getting nowhere! :)

Well, yeah. Axioms are not deduced, they are the starting points for the various theories that are deduced from them. (There is not much you can deduce ex nihilo.)

Did you have some further point?

If you were hoping that mathematics would have - or thinking that it _should_ have - some ontological foundation that is not in this sense "made up", I'm afraid it simply doesn't.

All of mathematics is built on sand in the sense that there necessarily are going to be undefined terms. There’s no way around this. Think about the English language. Grab a dictionary. Look up any word in it. Look at the definition of that word. Pick a word in that definition. Repeat and eventually you’ll end up with a defintition that contains one of the words you previously looked up. So English is built on sand too and yet we are able to still communicate.

> the natural 0 is the empty set.

That is a definition of "natural 0", not a definition of "empty set". GP was pointing out that GGP was using the concept of "empty set" to provide a definition for the natural numbers, without having first provided a definition for the empty set.

Replies that consist solely of throwing a quote at someone are kind of rude even if you're in the right.

I don't see why GGP should have to reinvent set theory to use it in his construction of numbers. Set theory and the idea of {} is well defined, and hardly "built on sand".

He doesn't necessarily have to. But when he defines "natural zero" as "the empty set", and someone asks "what's the empty set?", and you say "natural zero", you're being circular.

{} is the empty set, which as mentioned is natural 0.

GP understands the notation; he means to cast aspersions about the axiomatic project.

Fair enough, though there weren't really any "aspersions" cast. I'm aware you can unravel set theory if you care enough about that particular exercise. But, in context, you could also interpret that comment as being written by someone who isn't aware that the empty set is defined as {}.

Responding to someone's definition that natural 0 = {} that mathematics is a "castle built on sand" isn't exactly a cogent criticism of set theory.

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