Thread: Foundations of Mathematics

1)lndicate whether the following statements are true or false:

a)lt can not proved mathematically that 2+2=4

b)It can be proved that there is no paradox in mathematics.(give an example)

c)Hilbert proved that thereis a statement in matematics whicah is true but can not be proved

d) Gödel proved that there is a statement in mathematics which can neither be proved nor pisproved

3)What was the set that Russell used to show that there was a paradox in mathematics?

Thank you so much I am waiiting for your msg guys..

1.a) take a look at this link: 2p2e4 - Metamath Proof Explorer
b) the wording of this question is a little too vague, for my liking. if the statement is that: it can be proved mathematics is (logically) consistent, it is untrue. this isn't to say mathematics is inconsistent, it's just that there are important statements in mathematics that cannot be proven. an example: the Axiom of Choice, is it true or false? it's been shown that including AC and denying AC both lead to consistent versions of set theory.
c) my guess is you are studying Gödel's theorems. i decline to answer this question, and the following.

3) why, Russell's set, of course.

Originally Posted by Deveno

it's been shown that including AC and denying AC both lead to consistent versions of set theory.

It was my understanding that it is unknown whether or not ZFC is consistent. Did I make a mistake?

To add some things:

a)lt can not proved mathematically that 2+2=4

This is false. A particular proof depends on the axioms used.

b)It can be proved that there is no paradox in mathematics.(give an example)

I think the word "paradox" here is a synonym of "contradiction." A theory (including the whole of mathematics) is called consistent if one cannot prove a contradiction in it.

d) Gödel proved that there is a statement in mathematics which can neither be proved nor pisproved

Search for the First Gödel's Incompleteness Theorem.

3)What was the set that Russell used to show that there was a paradox in mathematics?

Search for Russell's paradox.

Originally Posted by roninpro

It was my understanding that it is unknown whether or not ZFC is consistent.

You are correct. From Wikipedia:

By work of Kurt Gödel and Paul Cohen, the axiom of choice is logically independent of ZF. This means that neither it nor its negation can be proven to be true in ZF, if ZF is consistent. Consequently, if ZF is consistent, then ZFC is consistent and ZF¬C is also consistent.

It's a bit strange that they would give
"b)It can be proved that there is no paradox in mathematics"
and
"3)What was the set that Russell used to show that there was a paradox in mathematics?"

"Mathematics" is a fluid concept...

Originally Posted by roninpro

It was my understanding that it is unknown whether or not ZFC is consistent. Did I make a mistake?

my bad, i should have said "consistent with ZF".