# Naked Science Forum

## On the Lighter Side => New Theories => Topic started by: sim on 11/01/2019 22:07:11

Title: Axiomatic set theory ZFC is inconsistent thus mathematics ends in contradiction
Post by: sim on 11/01/2019 22:07:11
Axiomatic set theory ZFC is inconsistent thus mathematics ends in contradiction

The meaninglessness of mathematics

Axiomatic set theory ZFC was in part developed to rid mathematics of its paradoxes such as Russell's paradox

The axiom in ZFC developed to do that, ad hoc,is the axiom of separation

Quote
. Axiom schema of specification (also called the axiom schema of separation or of restricted comprehension): If z is a set, and \phi\! is any property which may characterize the elements x of z, then there is a
subset y of z containing those x in z which satisfy the property. The "restriction" to z is necessary to avoid Russell's paradox and its variant

Now Russell's paradox is a famous example of an impredicative construction, namely the set of all sets which do not contain themselves

The axiom of separation is used to outlaw/block/ban impredicative statements like Russells paradox

but this axiom of separation is itself impredicative

Quote
"in ZF the fundamental source of impredicativity is the seperation axiom which asserts that for each well formed function p(x)of the language ZF the existence of the set x : x } a ^ p(x) for any set a Since the formular
p may contain quantifiers ranging over the supposed "totality" of all the sets this is impredicativity according to the VCP this impredicativity is given teeth by the axiom of infinity

but the axiom thus bans itself-thus ZFC is inconsistent

[axiom of separation] thus it outlaws/blocks/bans itself
thus ZFC contradicts itself and 1)ZFC is inconsistent 2) that the paradoxes it was meant to avoid are now still valid and thus mathematics is inconsistent
Which are now still valid

with all the paradoxes in maths returning mathematics now again ends in contradiction
Title: Re: Axiomatic set theory ZFC is inconsistent thus mathematics ends in contradiction
Post by: Kryptid on 11/01/2019 23:51:12
The meaninglessness of mathematics

If mathematics is meaningless, then why not donate all of your money to charity? If math is meaningless, then so is the amount of money you have.
Title: Re: Axiomatic set theory ZFC is inconsistent thus mathematics ends in contradiction
Post by: evan_au on 12/01/2019 08:57:24
It seems that you don't like self-referential statements?

But any system of communication sufficiently complex to be really useful will be able to make self-referential statements (like this forum post).

I think that self-reference is essential - even indirect cases such as the ability of your reply to refer to this post, which refers to the escape clause that would "outlaw/block/ban impredicative statements".