Comment by xamuel | Hacker News Reader

Comment by xamuel | Hacker News Reader

Article seems very verbose and only really addresses the natural numbers (it mentions negatives and rationals in passing).

In case anyone just wants actual answers without reading pages and pages of prose, here is one set of constructions (there are other competitors too).

The natural 0 is the emptyset. The natural 1 is the singleton {0}. The natural 2 is {0,1}. In general, the natural n+1 is {0,...,n}. Exercise to the reader: define appropriate arithmetical functions on the naturals.

Define a relation ~ on pairs (a,b) of naturals by saying (a,b)~(c,d) if and only if b+c=a+d. Exercise to the reader: ~ is an equivalence relation. Its equivalence classes are called "integers". The equivalence class containing (a,b) represents the integer b-a. For example, the integer -1 is the equivalence class {(1,0),(2,1),(3,2),...}. Exercise: define appropriate arithmetical functions on the integers.

Define a relation ~ on pairs (m,n) (n nonzero) of integers by saying (m,n)~(p,q) if and only if mq=pn. Exercise: Show ~ is an equivalence relation. Its equivalence classes are called "rationals". The equivalence class containing (m,n) represents the rational m/n. For example, 1/2 is the equivalence class {(1,2),(-1,-2),(2,4),(-2,-4),...}={(k,2k) for all nonzero integers k}. Exercise: define arithmetic operations on the rationals.

To get from the rationals to the reals, see https://en.wikipedia.org/wiki/Dedekind_cut

The thing that I don't like about this sort of construction is that it also implies all sorts of nonsense. For example, I can ask the question "Is 2 a member of 4", which is clearly nonsensical, but will get the answer "yes" from this model.

Type theory and category theory give us a much better way of constructing these sorts of objects without having to resort to creating constructions with all sorts of side effects. Not to mention that this is one possible construction of the natural numbers in set theory; in other formulations the theorem "is 2 a member of 4" would not be true, making things even more confusing. For example, we have the other standard approach of 0 = {}, 1 = {{}, 0}, n={{}, n-1}. The untyped lambda calculus and things like Church numerals have the same defect.

If we can start assigning types, then we can reason about these things in a much more concrete manner without these nasty side effects of theorems that only make sense because of definitional shortcuts. We can even derive isometries between different derivations of the same objects to show that two different definitions (like the Peano numerals and binary numbers) of a "natural number" can be used exactly interchangeably, because theorems like "is 2 a member of 4" are simply not expressible given the derivation.

> The thing that I don't like about this sort of construction is that it also implies all sorts of nonsense. For example, I can ask the question "Is 2 a member of 4", which is clearly nonsensical, but will get the answer "yes" from this model.

Why? Any set of 4 things also contains 2 things. Where's the nonsense? Within this definition, that's consistent.

> Type theory and category theory give us a much better way of constructing these sorts of objects without having to resort to creating constructions with all sorts of side effects.

Out of curiosity, what is the category theoretic construction that avoids Russell's paradox? I don't know that there isn't one, but I can't think of it off the top of my head. I know there are category theoretic constructions in general (I responded to one someone else posted in this thread).

The definition of the Peano numerals has two constructors, Zero: N and Succ: N -> N. Russell's paradox (assuming that we're talking about the idea of "set of all sets that are not members of themselves") is avoided simply because the objects produced are not sets, and sets have no exalted position within the mechanics of category theory.

Talking about concepts like "the category of categories that don't contain themselves" is kind of navel-gazey; and ends up falling apart in most constructive variants just because you can't give a comprehensive construction of elements of this category.

Admittedly I'm throwing some concepts from intuitionalism and type theory in the mix here; if I took the time I could make these statements more precise.

I see what you're getting at. My (similarly hand wavey) perspective is that constructions using category theory end up defining natural numbers as categories of sets with cardinality n. What I was really getting at is, how do you define your relation for the category in such a way that isn't pathological ("n is the category of sets with cardinality n except all such sets containing n")?

Alternatively, what are you selecting as the objects for your category if not sets of a given cardinality?

To be clear, I consider a lot of discussion about the foundations of set theory (and paradoxes thereof) to be pretty navel-gazey.

> Why? Any set of 4 things also contains 2 things. Where's the nonsense? Within this definition, that's consistent.

That's the problem -- it's only within this definition. In other definitions, notable the nesting example that I gave, that theorem is false. And the theorem makes no sense in and of itself, because we're talking about numbers, so it is unexpected that the membership operator would apply at all.

Whereas I can take two definitions of the natural numbers

  PN =
  Peano_One: PN
  Peano_Succ: PN -> PN

  BN =
  Binary_One: BN
  Binary_2x: BN -> BN
  Binary_2xp1: BN -> BN

and I can define a Plus: PN x PN -> PN and a Plus: BN x BN -> BN, and so on, and once I can define Binary_Succ: BN -> BN and Peano_2x and Peano_2xp1 I can prove that these are isometric types, so all theorems derivable from PN apply to BN and vice versa, not just the convenient theorems that don't use any syntax from the meta-language (e.g. set theory).

It's similar to the way in C you can malloc memory for two data structures, and then check which pointer is smaller. Say, one pointer points at the data for Mario, and the other pointer points at the data for Luigi. You look at the raw pointers and notice that the Mario pointer is a smaller 64-bit number than the Luigi pointer. That seems absurd or arbitrary because why should "Mario" be "smaller" than "Luigi"? But it doesn't really matter: the game still runs fine and nothing breaks as a result of the arbitrariness.

What you're saying isn't demonstrating any inconsistency. A thing is only true in mathematics if it follows from the definitions. If you change your definitions, you should expect that theorems built upon those definitions will no longer hold.

Stated another way, consistency is only a coherent mathematical concept from the perspective of a specific set of definitions. There's no problem here.

I don't claim that this makes it inconsistent. Just that it admits meaningless concepts. You can attribute this, if you like, to the fact that I am a computer scientist, and see things through that lens.

If you give me a statement saying "For two sets, A and B, is 'A union intersection B empty", the statement would not be well-formed, so you could just say that it is not a valid proposition without asserting anything about its truthiness. Similarly, a good foundation of mathematics should be able to reject a statement like "2 is a member of 4" as poorly formed, and make no statement about its truthiness.

I guess what I'm saying is that set theory is a bad basis for mathematics and should just be regarded as an interesting relic from previous attempts to formalize mathematics so that we can move on to more powerful systems.

I'm getting flashbacks of Baby Rudin's first chapter from this comment :)

For anyone who wants to learn more about this: what the parent comment is describing is the construction of natural numbers as cardinalities of sets. This is the (modified) Frege definition which avoids Russell's paradox - you can use that as a jumping off point for further dives into set theory and number systems.

To define Peano arithmetic you start with a successor function, then an addition function, then multiplication, etc.

”an equivalent relation.”

Auto-correct at work, I guess. For those not familiar with the term: if Google says did you mean equivalence relation, Google is right.

Thanks, fixed

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.

> 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.

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.

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