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I was reading Terence Tao's book on Real Analysis I. https://terrytao.wordpress.com/books/analysis-i/

My Background: I am not familiar with logic. And I am used to defining $\mathbb{N}$ in the context of set theory.

Question 1: How could Prof. Tao define $\mathbb{N}$ using Peano's axioms (Ch.2) before he defines sets (Ch.3)? What is the meta theory used to define $\mathbb{N}$

(From my understanding, a related post is this on foundations.) That is, he begins by simply saying :

Axiom 2.1: $0$ is a natural number. Axiom 2.2: if $n$ is a natural $n++$ is a natural number...

The logic seems to be as follows:

  • There is a world of mathematical objects
  • Natural numbers are such objects.
  • The $0$ is a type of mathematical object. The type is natural numbers.

Question 2: what are "objects" Prof Tao defines sets in chapter 3. In the book, he said the following:

enter image description here

Prof. Tao seems to use "objects" as a primitive notion, a notion that is to be assumed. Is this correct?

user26857
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Bryan Shih
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    At least, I don’t understand your question! What do you name a sort of foundation of mathematics? – mathcounterexamples.net Aug 20 '23 at 15:38
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    I don't have the book. Did he actually state the first two bullet points? He may be taking a category theoretical approach or this may be just pedagogy. This question needs a lot more context. Although many authors begin their books with a review of set theory, most of them use as much set theory as is necessary to develop their subjects. They don't try to develop every subject from scratch. – John Douma Aug 20 '23 at 15:53
  • @mathcounterexamples.net , as John Douma said, usually textbooks begin with set theory - which is what I am used to. But I am confused about the presentation. By "what source of foundation" I meant whether this is via presenting N as sets, or as a separate entity, e.g. Type Theory. Let me add some screen shots. – Bryan Shih Aug 20 '23 at 16:44
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    You don't need formal set theory or type theory to understand the logic of natural numbers that Tao is presenting. That's why the book "takes an agnostic position" on foundational technicalities - those choices don't make a difference here. – Karl Aug 20 '23 at 17:05
  • I don't know the context, but those axioms could fit in just predicate logic: $\operatorname{IsNatural}(0)$, $\forall n(\operatorname{IsNatural}(n) \rightarrow \operatorname{IsNatural}(n\mathord{+}\mathord{+}))$. Predicate logic itself is agnostic about the meanings of those things; assigning a meaning to each can be considered a model of the theory. – aschepler Aug 20 '23 at 17:08
  • @Karl may you elaborate/give any reference? I think this is just something I have always shrugged under the rug and never understood. Probably this is what aschepler is referring to? A post would be appreciated! – Bryan Shih Aug 20 '23 at 17:11
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    But that's just if you want or need to make everything precisely technical, and maybe he just doesn't consider that important at this point. If you dig into the logic behind proofs, when we prove things about those logical systems, we need to do that in a meta-logic; and eventually there's no way to precisely define it, it's just essentially "this is the way we think about logic". – aschepler Aug 20 '23 at 17:15
  • But now that it sounds more like
    1. Tao is defining the "properties of numbers" rather than explicitly constructing the "numbers". His construction doesn't explain numbers exist.

    2.I slightly updated my post - I am still confused what I'm confused, hope you can bare with me!

     – Bryan Shih Aug 20 '23 at 17:34
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    I think Tao expects to be read from a more naive/informal perspective. There doesn't need to be a notion of types or an explanation of what a mathematical "object" is. Just try to follow the logical arguments that he makes from the given axioms. The reasoning should make sense without depending on foundational choices. – Karl Aug 20 '23 at 17:43
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    Answering 1: Peano's axioms do not require notion of sets. He defines them by their properties. Whatever satisfies these properties is a natural number. He is not constructing them, because that's not important (as long as you know that they can be constructed). Answering 2: he refers to urelements and its theory. Rarely used nowadays, since everything can be done assuming that everything is a set. – freakish Aug 20 '23 at 18:10
  • Natural numbers and real numbers were studied and formalized before sets without mathematicians having a problem doing it, so just follow the historical path. Since the natural numbers have just one size, and the real numbers have just one size, it's easier to formalize them than formalizing all sets that want to have all sizes. – Chad K Aug 20 '23 at 19:44
  • @JohnDoe: Real numbers and sets were both formalised by Cantor (in fact, I think he introduced both in the same paper). Also, I wouldn't say that size is the singular issue that makes sets complicated. In fact, I would argue that the basic concept of a set is not really less elementary than the basic concept of a natural number, with the latter being an abstraction of finite collections. – tomasz Aug 20 '23 at 23:52
  • @tomasz historically tally-marks are the oldest mathematical example and they predate counting by hundreds of thousands of years. Shepherds would create a mark on the posts of their pen to match with their sheep. Each system of numeracy that was developed independently in world history use line like $1$ for one which is an artifact of tally-marks as well. So we seemed to understand set isomorphism before numeracy. – CyclotomicField Aug 25 '23 at 19:25
  • @CyclotomicField: I don't really understand what you're trying to get at. – tomasz Aug 27 '23 at 00:04

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The practice of much of modern mathematics is based on certain assumptions, such as the axiom of infinity (the existence of an infinite set). The practice (of over 100 years) shows that this does not get us into contradictions - though we can have no proof of that, by the incompleteness theorem. Historically, such practice has been challenged by philosophers. Already in the 1630s, Cavalieri's indivisibles were attacked by contemporary theologians/mathematicians. Cavalieri's response was that his critics were doing philosophy, not geometry.

To a certain extent, the mathematicians developed axiom systems so as to be able to counter such objections by making the assumptions explicit: these are our assumptions, and these are our conclusions. Practically speaking, if one can do without certain assumptions, it is arguably preferable to avoid them. This is the case with axiomatizing the natural numbers by means of Peano Arithmetic PA. Tao does it in such a way that the existence of $\mathbb N$ is not assumed; it is only a convenient piece of notation, as is the non-existent "set of all sets" in the traditional set theory ZF.

PA is weaker than ZF, and in particular has no provisions for infinite sets. To go further and start developing, for example, real analysis, it is convenient to use ZF (or more) as the background system. At that stage, it becomes necessary to explain how a model for PA can be constructed within ZF, which is what Tao does later in his book.

You mentioned in one of your comments that "Tao is defining the 'properties of numbers' rather than explicitly constructing the 'numbers'. His construction doesn't explain numbers exist." But Tao is not giving a construction of the natural numbers in the early part of the book; rather, he is presenting an informal axiomatisation of the natural numbers, along the lines of Peano Arithmetic. As mentioned above, this requires fewer foundational commitments than a set-theoretic framework such as ZF. You can't "construct numbers" without assuming anything.

Mikhail Katz
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  • this does not get us into contradictions - though we can have no proof of that, by the incompleteness theorem could you explain please, how does incompleteness theorem imply that we cannot know whether our assumptions lead to contradictions? – SBF Aug 21 '23 at 08:43
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    @SBF, if we cannot prove consistency of a particular framework, then we always have to live with the theoretical risk of contradiction. For instance, there may be an even stronger system (say $S$) than ZF that can prove the consistency of ZF, but then we can't have a proof of consistency of $S$ within $S$, by Goedel. Thus, trying to attain consistency would get us into infinite regress. – Mikhail Katz Aug 21 '23 at 08:52