Re: what makes it true?


>> You didn't ask for an explanation, you asked for a definition.

> A definition that makes sense to somebody who doesn't already
>know what is meant by the natural numbers.


What level of sophistication are you assuming then? If little
to none, then I can *only* give an intuition concerning the
natural numbers, not a definition. If you are assuming enough
sophistication to understand how to use the axioms of ZFC, then
my definition was perfectly good. I don't expect to be able to
give a definition of the set of natural numbers to someone who
doesn't have enough mathematical maturity to understand the
definition. I can elucidate the definition with an intuition,
but I don't expect the intuition to prevail when there's a
disagreement.

This reminds me of the difficulties giving a definition of the
term 'curve'. All of us have some intuition about what it means
to be a curve. But when it comes time to give a definition, it
becomes a lot stickier. If you define it as a continuous map
from [0,1], you violate the intuition that a curve is a subset
of, say, the plane. If you define a curve to set the image of such
a continuous function, you have the difficulty that the unit
square is a curve. If you require piecewise differentiability,
then you leave out the potential to talk about Brownian motion.
Which is the 'correct' definition? Whichever one we decide upon.
Whichever one captures what you need iot to capture.

For the natural numbers, our intuition is 0,1,2, and so on. The problem
is that phrase 'and so on'. What does it mean? The work around is to
adopt a definition in some axiom system that catches some of our
intuition. Since ZFC is usually used for mathematics, we define the
set of natural numbers as the smallest inductive set. That certainly
matches at least some of our intuitions about the natural numbers.
It also has the advantage that under fairly minor assumptions, any
set with a 'successor function' satisfying minimal properties will
be isomorphic to the natural numbers. This is essentially the uniqueness
of models of the 2nd order theory.

An alternative way to define the natural numbers is to use PA. What
is found is that models in ZFC of PA are not all isomorphic. In this
sense ZFC nails down the natural numbers more than PA does. That is one
reason the ZFC definition is my preferred one. Another is that I want to
agree on some axioms for mathematics and get down to business. ZFC does
most of what I need. PA doesn't. I actually prefer GB, but only because
classes are actual objects and category theory is easier.

--Dan Grubb
.

Relevant Pages

  • Re: Epistemology 201: The Science of Science
    ... there is no formula in the language of ZFC used in the proof. ... > formalizations of the theorems of mathematics. ... on intuition for proofs, because so many things defy intuition. ...
    (sci.math)
  • Re: Epistemology 201: The Science of Science
    ... there is no formula in the language of ZFC used in the proof. ... > formalizations of the theorems of mathematics. ... on intuition for proofs, because so many things defy intuition. ...
    (sci.cognitive)
  • Re: Epistemology 201: The Science of Science
    ... there is no formula in the language of ZFC used in the proof. ... > formalizations of the theorems of mathematics. ... on intuition for proofs, because so many things defy intuition. ...
    (sci.physics)
  • Re: ZFC
    ... >This might be formally true if, in addition, the question of the ... ZFC is still considered a foundation for mathematics regardless. ... It's true that if you insist to clinging to the intuition behind Hilbert's ...
    (comp.theory)
  • Re: Cantorian pseudomathematics
    ... you mean that mathematics should also incorporate handwaving and spurious assumptions based on one man's false intuition. ... , demands some work to define the naturals, more to define the reals, ... of limiting process to check that the number - the abstraction - sqrt ...
    (sci.math)
Loading