36
votes
Is there a paradox in the proof of Godel's incompleteness theorem?
Gödel was right. O'Connor 2005 meets every known objection: it is constructive and finite, indeed it runs on commodity hardware in reasonable time; it includes Rosser's trick, it is not relative to ...
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24
votes
Is there a paradox in the proof of Godel's incompleteness theorem?
You do not understand the incompleteness theorem. It does not require "coding", and it does not depend on "actual infinity", and it does not "hide" any paradox. You ...
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13
votes
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Gödel's incompleteness theorems - what are the religious implications?
Gödel's theism is discussed by Franzen in Gödel’s Theorem: An Incomplete Guideto Its Use and Abuse. He penned a version of the ontological argument, and in 1961 ranked the worldviews “according to the ...
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11
votes
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How does Gödel's incompleteness theorem apply to materialism and the mind
(Disclaimer: All of what follows is explicitly done under the assumption that the mind is a Turing machine and that we can formally axiomize our mathematical thinking. This has, of course, never been ...
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10
votes
Accepted
What did Gödel mean by "positive property" in his ontological argument?
"Positive" is what Leibniz and other proponents of the ontological argument called qualities that make something "better" than it is without them (Anselm spoke of "good" ...
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9
votes
Did Russell understand Gödel's incompleteness theorems?
Russell's comments on Gödel were scanty, but it was very unlikely that Russell did not understand what Gödel was talking about. The paradox presented by Gödel sentence was nothing new; it was the same ...
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8
votes
Why did Gödel believe that there was a conspiracy to suppress Leibniz's works?
The OP information probably comes from Wikipedia's article on Characteristica Universalis
"The logician Kurt Gödel, on the other hand, believed that the characteristica universalis was feasible, ...
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8
votes
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Does Gödel's argument that minds are more powerful than computers have the inconsistency loophole?
As such, Turing machines are not consistent or inconsistent - formalized systems are. However, once we fix a coding of (e.g.) the syntax of the language of arithmetic, we can view some Turing machines ...
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8
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Did Russell understand Gödel's incompleteness theorems?
As mentioned in a comment, Alasdair Urquhart has written a paper, Russell and Gödel (Bull. Symb. Logic 22 (2016), 504–520), that discusses a number of different topics, including Russell’s view of ...
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8
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Are axioms tautologies?
It is worth separating the logic from the epistemology. Let's start with the logic.
A (first order) theory is a set of sentences. Usually we are interested in deductive systems, so we require a ...
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8
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How to show (in a hand waving manner) that the Godel sentence is true
Gödel’s Incompleteness Theorem is a result about formal systems.
Its proof requires certain assumptions about the properties of specific formal system F: basically, about its "expressive capabilities"...
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7
votes
Gödel’s Incompleteness Theorem: How can truth go deeper than proof?
At least in the case of arithmetic, the concept of arithmetical truth can be defined as a type of non-computable extension of the Peano axiomatic system. Basically the idea is to consider the set of ...
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6
votes
Are axioms tautologies?
An axiom is something you assume to be true without proof. A tautology is a statement which can be proven to be true without relying on any axioms. An axiom is not a tautology because, to prove that ...
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6
votes
Gödel: Why is a proposition undecidable?
This answer is a bit technical, but I think the OP will find it interesting.
Let me first recall a bit about the history of the incompleteness theorem (IT). How IT is usually stated is:
If PA (or any ...
- 3,368
6
votes
Did Gödel do philosophy beyond logic?
The most oft-quoted of Gödel's philosophical works might be Is Mathematics a Syntax of Language? It is not available online, unfortunately, but excerpts are read in this YouTube video. Tait wrote a ...
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6
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Why is ZFC not as susceptible to Gödel's incompleteness as was the Principia Mathematica?
ZFC is susceptible to Gödel's incompleteness as was the Principia Mathematica.
See Gödel’s Incompleteness Theorems for an introduction to the theorem :
Any consistent formal system F within which ...
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6
votes
Accepted
Why can Goedel's Incompleteness Theorem be proven?
"Godel thus proved an unprovable statement" - this is not quite the case, as you yourself recognize in the lines following. Rather, Godel proved that there exists an undecidable statement ...
- 3,017
5
votes
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Is it valid to prove the axioms of a system from themselves? How does it square with Gödel's incompleteness?
Yes, the axioms do trivially prove themselves. Your last derivation, however, is not valid: "A=A" can not be substituted for A because the latter is a symbol in a formal system, while the ...
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5
votes
Can you list examples of problems that can not be solved within a formal system but human beings have solved through construction or creativity?
There is no agreed upon example of this kind.
Let us explore the issues:
First, we cannot even come up with a decent example of a problem not solvable by any formal system. If you state the problem ...
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The origin of a particular self-reference paradox
The origin is with the so-called Whiteley Sentence.
See C.Whiteley, “Minds, Machines and Gödel: A Reply to Mr. Lucas (1962)”, Philosophy 37:61-62 :
It is possible to devise a formula which will ...
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5
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Before Gödel, was undecidability of axiomatic systems an issue at all?
While it’s not quite a perfect parallel, a related concern about decision problems had been phrased some years before Gödel’s work: see https://en.m.wikipedia.org/wiki/Entscheidungsproblem . David ...
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4
votes
How does Gödel's incompleteness theorem apply to materialism and the mind
Assertion 1A: Humans use some finite logical system to understand the universe.
If you think our brains are equivalent to TMs, and believe Church-Turing, then infinity is going to be a challenge. But ...
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4
votes
How is ω-consistency different from ordinary consistency?
Yes, see ω-consistent theory. It plays a role in the study of formal theories in mathematical logic research, and is related to the so-called ω-logic needed to express the "true standard ...
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4
votes
How to show (in a hand waving manner) that the Godel sentence is true
(1) is a special case of the general principle that if you accept a statement, then you accept that the statement is true. If you believe snow is white, then you believe "Snow is white" is true. It ...
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4
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Why does Gödel's incompleteness theorem apply to multiple formal systems?
Short version:
No, this doesn't work.
First, note that you haven't stated Godel's theorem correctly - rather, it is:
Every consistent computably axiomatizable theory "containing enough ...
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4
votes
Accepted
Godel's incompleteness theorem when the cardinality of axioms is > ℵ_0?
Keeping things reasonably simple at the cost of some generality, Godel's (first) incompleteness theorem says the following:
There is no complete consistent computably axiomatizable theory in the ...
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4
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How is the completeness of first order logic reconciled with the incompleteness of set theory?
First Order Logic is complete in the sense that: there is a proof procedure for FOL such that just the statements(/wffs) of FOL that are true and remain true under any re-interpretation.
Yes; ...
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4
votes
Prerequisites for mathematical logic?
While mathematical logic is mathematical (it's right there in the name), as a principled targeted study, there is not much overlap with the usual mathematical undergraduate curriculum (at least in the ...
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