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Mathematics self-proves its own Consistency (contra Gödel et. al.)

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Re: Mathematics self-proves its own Consistency (contra Gödel et. al.)

#4

Post by GuiA »

Carl Hewitt is an interesting character [1] and I'm not really sure what to make of it. There are signs of a bright mind gone completely bonkers, and there are signs of someone being rejected for purely political (and not academic) reasons. Both have happened numerous times in the past which is why it's hard to tell. If someone more enlightened on the topic cared to comment about the nature of things, that'd be greatly appreciated :)

[1] http://www.guardian.co.uk/technology/2007/dec/09/wikipedia.i...

Re: Mathematics self-proves its own Consistency (contra Gödel et. al.)

#6

Post by acchow »

I quit after the first paragraph.

If you assume "proof by contradiction" to be a valid form, then you also assume that mathematics is consistent. Nothing to see here... Move along...

Edit: In case some people missed it... In our logical framework, an inconsistent theory can prove both Φ and ¬Φ. Notice that you are using our logical framework; you are assuming it to be consistent. Yes. Assuming the framework to be consistent, you can prove that the framework is consistent. Shocking.

Re: Mathematics self-proves its own Consistency (contra Gödel et. al.)

#7

Post by gphilip »

> both Φ and ¬Φ are theorems that can be used in the proof to produce an immediate contradiction. Therefore mathematics is consistent.

Isn't the inference that mathematics is inconsistent?

A proposition is a statement which is either true or false (and not both, and not neither).

An axiom is a proposition which we assume to be true, "for free".

A logical deduction is a way to combine or modify true statements to get other true statements.

A proof of a statement is a finite sequence of propositions. The sequence starts with an axiom, and ends with the target statement. Its other elements are propositions---these can be axioms or not. Each proposition which (i) is in the sequence and (ii) is not an axiom, must be the result of applying a logical deduction to one or more propositions which appear before it in the sequence.

A theorem is a proposition for which there exists a proof.

If we can prove two theorems which contradict one another (like Φ and ¬Φ in the quote above), then we say that our system for finding proofs is inconsistent. Since the OP starts out assuming that mathematics is inconsistent and ends up producing two contradictory theorems, he does not seem to get anywhere far. And his inference---that mathematics is consistent---seems to be incorrect.

What am I missing here?

Re: Mathematics self-proves its own Consistency (contra Gödel et. al.)

#8

Post by cobbal »

gphilip wrote: post #7

> both Φ and ¬Φ are theorems that can be used in the proof to produce an immediate contradiction. Therefore mathematics is consistent. Isn't the inference that mathematics is in consistent? A proposition is a statement which is either true or false (and not both, and not neither). An axiom is a proposition which we assume to be true, "for free". A logical deduction is a way to combine or modify true statements to get…

The beginning of the proof starts with "Suppose to obtain a contradiction, that mathematics is inconsistent".

From that assumption, they derived a contradiction, therefore the assumption can't be true, and math must be consistent.

Re: Mathematics self-proves its own Consistency (contra Gödel et. al.)

#9

Post by jeffcoat »

acchow wrote: post #6

I quit after the first paragraph. If you assume "proof by contradiction" to be a valid form, then you also assume that mathematics is consistent. Nothing to see here... Move along... Edit: In case some people missed it... In our logical framework, an inconsistent theory can prove both Φ and ¬Φ. Notice that you are using our logical framework; you are assuming it to be consistent. Yes. Assuming the framework to be con…

Did you see the first line? "That mathematics is thought to be consistent justifies the use of Proof by Contradiction." So the original author agrees with you.

You're right too, though: most of the value of that post is in the first paragraph, which is a very concise presentation of an interesting (apparent) paradox.

Re: Mathematics self-proves its own Consistency (contra Gödel et. al.)

#10

Post by gphilip »

cobbal wrote: post #8
gphilip wrote: post #7

> both Φ and ¬Φ are theorems that can be used in the proof to produce an immediate contradiction. Therefore mathematics is consistent. Isn't the inference that mathematics is in consistent? A proposition is a statement which is either true or false (and not both, and not neither). An axiom is a proposition which we assume to be true, "for free". A logical deduction is a way to combine or modify true statements to get…

The beginning of the proof starts with "Suppose to obtain a contradiction, that mathematics is inconsistent". From that assumption, they derived a contradiction, therefore the assumption can't be true, and math must be consistent.

> From that assumption, they derived a contradiction

This is what I don't see. To have derived a contradiction, they should have shown: "mathematics is consistent". Instead they ended up showing (because Φ and ¬Φ): "mathematics is inconsistent". Which is what they started out with. Where is the contradiction?

Hmm.. while writing this out, I think I am starting to get the source of the confusion. "Mathematics is inconsistent" is a meta statement about mathematics, and "Φ and ¬Φ" is a "plain" statement within the said mathematics. My intuitive feeling is that the latter contradiction does not "affect" the meta statement. I need to think further to make this more rigorous.

OT: Why the downvote(s)? Does my comment take away from the discussion?

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