Thursday, April 04, 2019

Kurt Gödel's Incompleteness Theorems (1931): Paraphrased

Kurt Friedrich Gödel

My Paraphrase of Kurt Gödel's Two Incompleteness Theorems

These are theorems of mathematical logic showing the intrinsic limitations of any formal axiomatic system that can model basic arithmetic.
The first states that no consistent system of axioms whose theorems can be listed by an algorithm can prove all truths about the arithmetic of the natural numbers. For any such consistent formal system, there will always be statements about the natural numbers that are true, but that are unprovable within the system.

The second incompleteness theorem extends the first by showing that the system cannot reveal its own consistency.
I hope I have stated these theorems correctly. Here's a roughly expressed example:
"This statement cannot be expressed within the Russell-Whitehead formal system set forth in Principia Mathematica."
Note the paradox. The statement is false if it can be expressed within the Russell-Whitehead system, and the statement is true if it cannot be expressed within the Russell-Whitehead system. Consequently, the Russell-Whitehead system is either incoherent or incomplete.



At 10:52 AM, Blogger Carter Kaplan said...

Did Gödel write this in response to something a Positivist said?

And it is Philosophy of Mathematics. Not my field. I am interested in the sense of propositions: True or false, the case or not the case, or appropriate or not appropriate, with emphasis on this last, third, concern.

Here is an intro to my methodology and interest, and it also reflects the sorts of thongs I am looking for in literature (or putting into my own fiction):

The Synpotic Surview.

As a methodology, the "Synoptic surview" is the effort of telling stories about the way words and concepts are used and understood.

See also the epigraph to Emanations 6 (the quote from Wittgenstein's On Certainty).

At 11:22 AM, Blogger Carter Kaplan said...

Hmm, back to Gödel:

I'd like to have a look at Wittgenstein's remarks on the theorem. Especially the "Notorious Paragraph."

Could be Wittgenstein nailed it? Or was he mistaken, as some have said? (Good question for Peter Hacker).

At 11:44 AM, Blogger Horace Jeffery Hodges said...

Wittgenstein first aligned himself with Russell and Whitehead. They wanted to reduce all math to logic. But Wittgenstein moved quickly away from their system, and I'm pretty sure he accepted Goedel's proofs. Even Russell and Whitehead themselves did so.

Jeffery Hodges

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At 1:53 PM, Blogger Horace Jeffery Hodges said...

Oh, I just now looked at the notorious paragraph. Don't listen to me . . .

Jeffery Hodges

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At 2:02 AM, Blogger Carter Kaplan said...

What does W say? Do you have a link?

At 6:12 AM, Blogger Horace Jeffery Hodges said...

I followed your link to this place this place.

Jeffery Hodges

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