- #1

- 2,544

- 3

The number of theorems that we could have is countable. I was just wondering what we might be able to say about how much we could know about math.

You are using an out of date browser. It may not display this or other websites correctly.

You should upgrade or use an alternative browser.

You should upgrade or use an alternative browser.

- Thread starter cragar
- Start date

- #1

- 2,544

- 3

The number of theorems that we could have is countable. I was just wondering what we might be able to say about how much we could know about math.

- #2

- 795

- 7

The number of theorems that we could have is countable. I was just wondering what we might be able to say about how much we could know about math.

A proof of a theorem is a

The union of the proofs of length n as n ranges over 1, 2, ... is a countable union of countable sets, so it's countable.

In other words, a countably infinite number of axioms doesn't buy you any more math than a finite set of axioms.

- #3

- 2,544

- 3

Ok I see what you are saying. So there is no way to derive an uncountable number of statements from a countable set.

- #4

chiro

Science Advisor

- 4,790

- 132

One thing I have to ask is whether the axioms are explicitly defined or not.

It may sound like a stupid question, but if you can define an axiom non-explicitly then you might have a different kind of case to work with as opposed to defining them all explicitly.

So what I mean is that by explicit you have an explicit definition for the axioms and then as SteveL27 said, you generate all possible statements as derived from those axioms (i.e. the rest of the axioms are 'unpacked' from the definition of the minimal set).

In this context the language used to define the axioms are explicit since the axiomatic definitions can not change. In an implicit context, this doesn't hold.

- #5

Bacle2

Science Advisor

- 1,089

- 10

Ok I see what you are saying. So there is no way to derive an uncountable number of statements from a countable set.

An uncountable number of statements, yes, but not an infinite number of proofs, if

you consider a proof to be a finite collection of statements, as SteveL pointed out. Sorry if this is what you

meant--good question, BTW.

- #6

Stephen Tashi

Science Advisor

- 7,642

- 1,495

good question, BTW.

Yes!

If we let our minds race we can daydream about Dedekind cuts for proofs. We'd need to define an order relation on proofs and have an axiom that says "The limit of any bounded monotonic sequence of finite proofs is a proof".

Share: