Can someone please help me understand the following assertion:

All concrete mathematics of the past can be conducted in Peano Arithmetic.

This is from "A Brief Introduction to Unprovability" by Andrey Bovykin.

Bovykin says that theorems of $\mathsf{PA}$ capture ‘finite mathematics’, that is the world of mathematical theorems that can be formulated in $\mathcal L = \{+, \times,< ,0,1\}$ and whose proof does not require the use of any notion of ‘infinite set’ in an essential way.

He says ‘finite mathematics’ includes "all imaginable mathematics whose objects can be somehow finitely approximated or finitely encoded, including everyday ‘continuous’ mathematics".

I understand how $\mathsf{ZFC}$ can be a framework for `everyday mathematics' but I don't see how $\mathsf{PA}$ can be. For example, how can I even state the l.u.b. property of the reals using $\mathsf{PA}$?

  • $\begingroup$ PA? Please define your terms. Try to write questions that are as good, clear, and findable via search as possible. $\endgroup$
    – Newb
    Jul 5 '14 at 5:44
  • $\begingroup$ For a large part of Number Theory, and stuff that can be encoded using Number Theory (much of discrete mathematics), first-order PA is powerful enough. Roughly speaking it is as powerful as the theory of hereditarily finite sets. $\endgroup$ Jul 5 '14 at 5:46
  • 1
    $\begingroup$ @Newb: I updated my question. I hope it looks better now. $\endgroup$
    – user2484
    Jul 5 '14 at 15:07
  • $\begingroup$ @user672484 Indeed --- now, this is an excellent question. $\endgroup$
    – Newb
    Jul 5 '14 at 18:00

The point is that one can use a lot of coding to reduce a great deal of fancy mathematics to Peano arithmetic, and even weaker theories. But to really get things going, you would want to use second-order arithmetic, or at least a subsystem of it. Have a look at the work on Reverse Mathematics, where such things are studied systematically.

In essence, "finite discrete mathematics" such as number theory, graph theory, basic combinatorics, etc., cane be reduced to PA because finite discrete objects can be encoded by numbers. Infinite objects, such as real numbers, continuous functions, distributions, etc., require a stronger theory, and second-order arithmetic suffices for a lot of these.


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