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31 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 ...
Corbin's user avatar
  • 1,547
27 votes

What's the solution to Sorites paradox?

The solution is to realise that the problem as posed is based on a false assumption that there is always a clear dividing line between two opposing classifications of degree. Take long and short, ...
Marco Ocram's user avatar
  • 23.5k
26 votes

What is the meat paradox?

There is no necessity for a solution because there is no paradox. This falsely called "paradox" is just a misinterpreted and biased statement, sustained by fallacies and political arguments: ...
RodolfoAP's user avatar
  • 7,661
22 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 ...
user21820's user avatar
  • 686
19 votes

What framework or tool solves the Barber Paradox?

It is worth noting that although the Barber paradox is often compared with Russell's paradox, it does not require any set theory. It can be formulated in elementary logic as follows: (∃x)(∀y)(Person(x)...
Bumble's user avatar
  • 26.3k
15 votes

Why would this not resolve the Sorites paradox?

Your proposed solution does not solve the paradox. The whole point of the paradox is that the term 'pile' is vague. That is, given an object (e.g. a collection of grains of sand) it is indeterminate ...
E...'s user avatar
  • 6,556
12 votes
Accepted

What the Preface paradox tells us about the principle of explosion

I agree with Just Some Old Man's answer, but to expand on it a little... If we think of all the statements in the textbook as propositions A1, A2, ..., An then the situation we are trying to describe ...
Bumble's user avatar
  • 26.3k
12 votes

Is there a distance so small it can't be further divided?

Is there a distance so small it can’t be further divided? The modern solution to this problem is the use of infinitesimals, as used by Leibniz and Newton in their development of the calculus. ...
Mark Andrews's user avatar
  • 6,314
12 votes

Is there a distance so small it can't be further divided?

The time it takes for the arrow to reach one half of the distance, is one half of the time. So the total length traveled by the arrow is one half the distance, plus one half of one half, plus one half ...
Stef's user avatar
  • 970
11 votes
Accepted

What's the current status of the "paradox of analysis"? And are there any strong and widely accepted resolutions?

A good paper to read on this subject is an old classic: Gilbert Ryle's Systematically Misleading Expressions. (Proceedings of the Aristotelian Society, 32: 139-170 (1932). Also in his Collected Papers,...
Bumble's user avatar
  • 26.3k
11 votes
Accepted

Is a set containing itself already a paradox?

Russell's paradox arises within naïve set theory by considering the set of all sets that are not members of themselves. Such a set appears to be a member of itself if and only if it is not a member of ...
Mauro ALLEGRANZA's user avatar
11 votes

What qualifies as the solving of a paradox?

Technically speaking, a paradox isn't a problem to be solved in and of itself. A paradox points at a weakness, misconception, or internal contradiction of the philosophical/analytical structures we ...
Ted Wrigley's user avatar
  • 20.4k
11 votes

Irresistible force paradox as an inconsistent formal theory

This dilemma doesn't really makes sense in modern physics because any macroscopic force accelerates any mass no matter how large, but if you were to axiomatize it in pre-modern physics then yes, it ...
David Gudeman's user avatar
11 votes

What is the meat paradox?

I'm joining the other answers in stating that the so-called "meat paradox" isn't a paradox. Rather, it's an example of the logical fallacy of all or nothing thinking. I strongly disagree ...
Syntax Junkie's user avatar
11 votes

What's the solution to Sorites paradox?

One solution would be to say that even 1 grain is a heap. That would be defining "heap" more precisely than its informal, intuitive meaning. What does "heap" precisely mean anyway? ...
Frank's user avatar
  • 2,454
11 votes

What's the solution to Sorites paradox?

The answer is in the definition of a heap. I offer the definition that a heap must have at least one layer stacked upon a base layer. For grains of sand you need at least 3 grains in a base layer to ...
Xavier's user avatar
  • 211
10 votes

Why would this not resolve the Sorites paradox?

This is all about the difference between natural language and formal language. In formal language, a term cannot be used unless it's well-defined according to the standards of the language. In ...
Chris Sunami's user avatar
  • 30.3k
10 votes

Is finding truth possible?

You've stumbled upon an old problem in philosophy, The Paradox of Inquiry, first formulated in Plato's Meno. The problem can be reformulated as follows: Either you know the answer to a question, or ...
E...'s user avatar
  • 6,556
10 votes
Accepted

What is the meat paradox?

Humans are irrational. There are countless examples of individual behaviour that shows inconsistency of beliefs. Christians condemn murder, but many will quite happily support the idea of war. People ...
Marco Ocram's user avatar
  • 23.5k
9 votes

What is the meaning of “to fight a Catch-22 is to accept it”?

The definitive way to answer the question is to read the book. Here is how it is presented in the book itself (copied from here): Yossarian: “You mean there’s a catch?” “Sure there’s a catch,” Doc ...
Roger V.'s user avatar
  • 1,565
9 votes

What if all sets contain themselves?

I think that disallowing the empty set will cause the resulting theory to lose too much. For example, the intersection of two disjoint sets, which would be the empty set, is now undefined. You will ...
Frank's user avatar
  • 2,454
9 votes

What if all sets contain themselves?

The axiom of foundation is not (usually) taken for a logical truth (although see about predicativism for an attempt to situate well-foundedness on the level of predication theory). Accordingly, its ...
Kristian Berry's user avatar
9 votes
Accepted

What if all sets contain themselves?

I think we can find a contradiction with ZFC immediately: Let S = {A, B} with A =/= B. WLOG, by your hypothesis, assume that A = S. (One of these elements needs to equal the whole set!) Then, also ...
waf9000's user avatar
  • 114
8 votes
Accepted

Why would this not resolve the Sorites paradox?

The previous answers betray a lack of familiarity with the literature. Your solution, using the least number principle, essentially works. It is a known argument for epistemism about vagueness, the ...
windlessq hickory's user avatar
8 votes
Accepted

Can it be rational to have beliefs one knows to be inconsistent?

It can. Ramsey put it nicely in his "last papers" written around 1929 under the influence of Peirce's pragmatism (quoted from Marion, Wittgenstein, Ramsey and British Pragmatism): "We want our ...
Conifold's user avatar
  • 43.5k
8 votes

What is the meaning of “to fight a Catch-22 is to accept it”?

By fighting the rule, you are accepting that it is something you have to deal with, whether obeying, evading, or fighting. If, for instance, soldiers were allowed to leave base before 9 with permits ...
Mary's user avatar
  • 1,988
7 votes

How do mathematicians reconcile that an infinite set does not have to be larger than its proper subset?

There is a long controversy as to what should count as the "size" of an infinite set, and there provably does not exist a notion that satisfies both the bijectivity principle, a.k.a. Hume's ...
Conifold's user avatar
  • 43.5k
7 votes
Accepted

Is it possible that a question has only two answers?

Answer 1 - 'yes' Suppose I ask, 'Can Tom walk ?' I am asking about the truth-value of the proposition, 'Tom can walk'. I expect the answer 'Yes' or 'No'. The answer, 'Yes', is right if Tom can walk ...
Geoffrey Thomas's user avatar
  • 35.8k
7 votes

What's the solution to Sorites paradox?

So I would say that two possible answers to the paradox are Rigorously define a 'heap' to explicitly consider the number of grains of sand, or Keep the fuzziness around heaps, and observe that tiny ...
Bug Catcher Nakata's user avatar

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