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