If we can know God perfectly, we can prove everything in mathematics once we fully know him and Godel's theorem will not apply. — Gregory

If you were to know God perfectly, mathematics is irrelevant. Especially if that "everything" includes things that mere mortals have deemed false. Just relax and savor the bliss. :love:

it is a theorem of first order logic that there is not an x such that for all y, y bears relation R to x if and only if y does not bear relation R to y. — TonesInDeepFreeze

Is this what you were referring too? It is not math but philosophy. There is nothing A unless B does not have a relation with itself? This what I'm talking about. Godel loops his arguments. Math and logic are different disciplines and combining them is a questionable enterprise. On the ladder of knowledge 1 plus 1 equaling two seems to come before the logicism used to prove this by Russell and Whitehead. So maybe their 700 pages on this is nonsense, a putting of the prior after what should come latter. And maybe Godel's ideas have the same problem: too much application of logic to math

Is this what you were referring too? — Gregory

I told you twice what Godel's theorem is:

If a theory T is a consistent, recursively axiomatizable extension of Robinson arithmetic, then there is a sentence G in the language for T such that neither G nor ~G is a theorem of T. — TonesInDeepFreeze

Instead of recognizing that, you bring up a different matter.

You will not make any progress here if you can't pay attention.

I told you what I thought of it. It does not mean anything mathematically because it refers back on itself, a move of logic, not math.

If a barber shaves only those who do not shave themselves then the barber doesn't shave himself. That's obvious. If it's stated in more complex terms, it is confusing two concepts as being one and needs normal language to clarify. That a professor of mathematics (Frege) got tripped up by this shows how poorly thought out his program was

That a professor of mathematics (Frege) got tripped up by this shows how poorly thought out his program was — Gregory

What “tripped up” Frege was Russell's paradox, not the barber paradox.

What “tripped up” Frege was Russell's paradox, not the barber paradox. — Amalac

Russel said he took the squiggly part of the barber paradox and used it with sets

Moves I've made by Hegel in philosophy have been applied to mathematics where they probably don't don't belong. The logical empiricists were stuck on things like "the black raven paradox" because they couldn't figure it out in their language. Russell himself said that after writing Principia Mathematica his mind was unclear on other subjects for many years latter. When you study one subject, it is supposed to increased your prowess in others.

Wrong, here's what Russell actually did say about the barber paradox:

You can modify its form; some forms of modification are valid and some are not. I once had a form suggested to me which was not valid, namely the question whether the barber shaves himself or not. You can define the barber as "one who shaves all those, and those only, who do not shave themselves". The question is, does the barber shave himself? In this form the contradiction is not very difficult to solve. But in our previous form I think it is clear that you can only get around it by observing that the whole question whether a class is or is not a member of itself is nonsense, i.e. that no class either is or is not a member of itself, and that it is not even true to say that, because the whole form of (the?) words is just noise without meaning. — Russell

The source is “The Philosophy of Logical Atomism”.

I told you what I thought of it. It does not mean anything mathematically because it refers back on itself, a move of logic, not math. — Gregory

Do you know about Gödel numbering? If so, you should be able to understand why the Gödel statement is a mathematical statement.

Godel and Russell both had many ideas that were mathematical but had an element of the science of logic in how they move. What Russell said in your quote is what I was saying. Sets that contain themselves are not objects of mathematics

Does the set of all sets that do not contain themselves contain itself. If we say no it is because you get put the set inside itself. What is in the set is too different from the set. Now we can say yes in that the set could contain it's items in itself and itself as well be in a different way

The Gödel statement is not a set, it's a statement.

It references itself, but unlike other statements that can be classified as meaningless, like the liar statement, it must either be true or false, because it's a mathematical statement.

Here you have a more detailed explanation

I told you what I thought of it. — Gregory

What you think about it is one matter. (What you think about it is based on a collection of confusions and misunderstandings you have.)

But you said that I have difficulty stating Godel's theorem. Yet I stated it without difficulty. So I'd like to know whether you recognize that you were incorrect to claim that I have difficulty stating the theorem. Your recognition of that would help to show that you are not entirely irrational and rhetorically irresponsible.

Russell's paradox is interesting philosophically, but I showed how this paradox can give two answers (both "in a sense"). Everyone has different explanation on how Gödel's arguments are supposed to work, probably from the nature of the case. If you disagree on logic's relation to math, then start with what you think are the logical tools of Gödel's theorem

You provided a statement and have not spoken yet of the internal logic that makes it a proof

If you disagree on logic's relation to math, then start with what you think are the logical tools of Gödel's theorem — Gregory

You only need these tools: knowledge about Gödel numbering, knowledge about formal systems, and the law of the excluded middle.

Did you even bother reading the Wikipedia article? It's not that long considering the difficulty of the subject.

Anyway, here is just one part of it:

Gödel noted that statements within a system can be represented by natural numbers. The significance of this was that properties of statements – such as their truth and falsehood – would be equivalent to determining whether their Gödel numbers had certain properties. The numbers involved might be very long indeed (in terms of number of digits), but this is not a barrier; all that matters is that we can show such numbers can be constructed.

In simple terms, we devise a method by which every formula or statement that can be formulated in our system gets a unique number, in such a way that we can mechanically convert back and forth between formulas and Gödel numbers. Clearly there are many ways this can be done. Given any statement, the number it is converted to is known as its Gödel number. — Wikipedia (Gödel numbering)

If the Gödel statement were meaningless, then we would not be able to construct it through Gödel numbers.
The Gödel statement is a meaningful statement since its corresponding Gödel numbers can be constructed. Therefore, it's either true or false, in the same way as a statement such as “10001^26278283 is prime” is either true or false, but not meaningless:

The Gödel sentence is designed to refer, indirectly, to itself. The sentence states that, when a particular sequence of steps is used to construct another sentence, that constructed sentence will not be provable in F. However, the sequence of steps is such that the constructed sentence turns out to be GF itself. In this way, the Gödel sentence GF indirectly states its own unprovability within F.

To prove the first incompleteness theorem, Gödel demonstrated that the notion of provability within a system could be expressed purely in terms of arithmetical functions that operate on Gödel numbers of sentences of the system. Therefore, the system, which can prove certain facts about numbers, can also indirectly prove facts about its own statements, provided that it is effectively generated. Questions about the provability of statements within the system are represented as questions about the arithmetical properties of numbers themselves, which would be decidable by the system if it were complete.

Thus, although the Gödel sentence refers indirectly to sentences of the system F, when read as an arithmetical statement the Gödel sentence directly refers only to natural numbers. It asserts that no natural number has a particular property, where that property is given by a primitive recursive relation. As such, the Gödel sentence can be written in the language of arithmetic with a simple syntactic form(...) — Wikipedia (Gödel incompleteness theorems)

That's a (very) brief summary of it.

If you want to know more, look it up.

I already have read that. The numbers are random and don't form real equation. I was asking for Gödel's theorems stated as verbal paradoxes like Russell's paradox. That way I can explain like I can do with Russell's. Maybe Gödel proves something but it's only about human cognition. The point of my thread was that higher species know things in better ways and spirituality can lead to thinking beyond human thought. How this works with Gödel's theorems is what I was wanting to talk about

You provided a statement and have not spoken yet of the internal logic that makes it a proof — Gregory

You said that I have difficulty stating the theorem. Stating a proof of the theorem is more than stating the theorem.

I stated the theorem without difficulty.

As I said, I'd like to know whether you recognize that you were incorrect to claim that I have difficulty stating the theorem. Your recognition of that would help to show that you are not entirely irrational and rhetorically irresponsible.

the law of the excluded middle. — Amalac

The incompleteness proof is intuitionistically valid and does not require excluded middle.

Hmm, ok. I meant that you may need the excluded middle (as it applies to mathematical statements) to show that the Gödel sentence was not meaningless despite being self-referential, like the sentence “this sentence is false”.

I thought the law of the excluded middle was also needed for mathematical proofs by contradiction, like Euclid's proof that there are infinitely many primes.

properties of statements – such as their truth and falsehood – would be equivalent to determining whether their Gödel numbers had certain properties. — Wikipedia (Gödel numbering)

That doesn't sound right to me. The properties that are checked are syntactical. What semantical properties does Wikipedia claim are checked?

Godel's numbering might not apply to the real world. The real world is mathematical but there might be a theory of everything in term of physics. If there are things we can't prove in mathematics, we at least knows math is true for us. For some Christians God gives us actual grace for our free will to renovate itself and return to it's former pure state and the merits of Jesus make us innocent before the justice of God. So man becomes somewhat divine Lutherans from Jacob Boehme to Hegel emphasized that God was "all in all", or in a more exact sense, was "everything in everything". All religions describe a union with the divine and if our thoughts can be raised up and the experience of the divine and is intellectual in a sense, it's possible our thoughts can be moved where everything is seen as a total unity of truth. This was my concern with Godel. Perhaps Godel helps us gain this vision, idn

I thought the law of the excluded middle was also needed for mathematical proofs by contradiction, like Euclid's proof that there are infinitely many primes. — Amalac

No, there are two kinds of proof involving contradiction:

assume ~P
derive contraction
conclude P

assume P
derive contradiction
conclude ~P

The first one requires excluded middle (or double negation, or whatever intuitionistically invalid rule).

The second one does not require excluded middle (or double negation or any intuitionistically invalid rule).

I see, that does make sense I guess.

Gödel sentence: I am not provable. Each unprovable yet true theorem is Göd.

I always enjoy reading logical ping pong, it's entertaining.
Paradox is just a logic hiccup. The Universe still gets on with doing what it does, despite human musings about paradox and/or infinities.
I am always surprised that an 'intelligent' person can ever satisfy their personal search for the T.O.E,
with the god fable. Especially when it clearly is just a filler story (or god of the gaps) for phenomena that humans just can't explain yet. As has already been suggested, it is lazy thinking.

For me, despite the barber's paradox, everyone that wants a shave, seems to be able to get one, despite any protest from, or hiccup in, propositional logic.
Paradox and infinities just indicate that we have not answered all the questions yet.
I hope we never do, as I am not sure what we would be for after that.
If we do ever answer every question, it will perhaps turn out that the Universe will become 'self-aware' and all sentient life in the Universe, can as a totality of thought, declare itself God.
Because at that point, the totality of thought from all life in the Universe would effectively satisfy the three qualifiers for godhood,
Omnipotence (no more questions to answer so this must have been achieved),
Omnipresence (no part of the Universe would exist, which is not affected by the totality of life within it), and Omniscience (ditto with omnipotence)

As long as unanswered questions exist, there is no God.
If all questions have been answered then God exists as a totality of life in the Universe.
Universeness 28/01/2022 TPF

ha ha...... :naughty:

Each unprovable yet true theorem is Göd. — Agent Smith

"What have I got in my pocket?" :wink:

"What have I got in my pocket?" :worry: — 180 Proof

Bad hobbit, Bad!