The issue of the type of a Goedel number apparently is trying to get buried. Again, consider the type of a Goedel number G that stands in place of a propositional variable. If G has type 'constant', then it doesn't have type 'variable', and thus can't be a propositional variable. If G has type 'variable', then it doesn't have type 'constant', and since all numbers have type 'constant', it's not a number. Thus, the concept of a Goedel number leads to a contradiction.

The above shows that the concept of a Goedel number is nonsense. Ignoring it, or trying to bury it from view, doesn't make the above any less valid.

Noah Schweber said "Or, treat it this way: consider the map that takes a string - with some symbols $a$ and $b$ that we think of as variables - and spits out the string gotten by replacing every "a" with "0" and every "b" with "S(0)". " Yes, there's a problem if you replace every variable with a constant. It takes what has possibility of being more than one thing and reduces it down to something that can only be one thing. One does not equal two, nor does two equal one!

And before someone says "but "0" is a symbol not a number" the problem here lies in that if the encoding of formulas involved just numerals, then using arithmetical theorems is not correct, and arithmetical theorems DO get used in that context. After all, arithmetical theorems apply to numbers, not to numerals.

The above shows that the Godel denials are nonsense. Failure to understand Godel's theorems does not make them any less valid.

Right... no one has a refutation that Goedel number entails the contradiction of an entity that is both a constant and a variable. They would rather bury that problem from view.

@DougSpoonwood: For the last time, if you are serious about learning, go get yourself a proper teacher, since you apparently cannot learn from textbooks that you said you acquired 4 years ago...

4 years is a really really long time to be unable to learn something, you know.

If you at least are willing to admit you don't understand what Godel was doing, a lot of experts here may be willing to actually sit down and explain to you.

@user21820 Where did I say anything about acquiring textbooks 4 years ago? And you aren't serious about learning, since you're trying to bury an issue.

Goedel was, in part, making a scheme that attempted to make numbers into both constants and variables.

See above comment made by you nearly 4 years ago.

On the same misconception about Godel's proofs.

@DougSpoonwood what's a variable?

@user21820 That book didn't address the issue as I recall.

@DougSpoonwood That's why I said... "you apparently cannot learn from textbooks that you said you acquired 4 years ago".

It's fine if that's the case. Just admit that you can't understand it.

@AlexanderGruber A variable is something that stands for more than one possible thing.

Maybe the textbooks are not written that good. Who knows.

@DougSpoonwood And that's where you start to go wrong already.

@DougSpoonwood I don't understand what that means. Define that.

@user21820 That book wasn't a textbook. It was more of a report.

@DougSpoonwood Then I have a recommendation for you. Rautenberg's "A concise introduction to mathematical logic".

@user21820 Oh really? Are you going to argue that a variable can not vary?

@DougSpoonwood I didn't argue anything of the sort that you seem to want to stick into my mouth even though I don't like the taste.

@user21820 Still haven't addressed the issue. Just handwaving.

Just answer Alexander's question.

I answered his question, he last gave me a request to 'define that'.

Yeah, define it please.

Since he gave me a definition, he wants me to define a particular definition.

No, I'm not looking for a particular definition that I know already.

I don't know much logic, Doug. You're gonna have to help me out here.

I don't think that definitions don't have definitions themselves.

Strings in propositional logic are not entirely meaningless... they have some properties... like variables being able to stand for more than one thing.

What does it mean for a variable to stand for more than one thing?

@AlexanderGruber It means that the number of things that can get put in its place is at least two.

And that the variable is not a constant.

Numbers are constant, and thus not variables.

And numbers have the type 'constant'.

Well, for example, if you talk about functions in calculus that have x in them, what those functions represent is a relation, which is a subset of the cartesian product of two sets

So, those functions are a set, and the x variable is a shorthand way of representing what that set is

Is that the sense in which you're using the word variable?

No, that's not the sense that I'm using the word 'variable', since functions like f:0->0 exist in that context. 'x' there would just be another notation for '0' only, and thus 'x' could take on only one value. The 'x' in (x+4)=2 is also not a variable, it's the constant '-2'.

@DougSpoonwood to define the function f you're talking about, the constant function, you need a domain and a codomain. f:A->B with f(x)=0. But this represents a subset of AxB defined by {(x,0) : x in A}. That isn't the same thing as the number 0.

Yes, the function isn't the same thing as the number 0. However, what is 'x'? Can it be anything other than the number 0?

@DougSpoonwood the notation "f(x)=0" is just a way of defining the set {(x,0) : x in A}

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@DougSpoonwood Let's continue here

So in this case the x itself has no meaning without the rest of the expression

An algebraist might say something about how you can define x as as part of an extension ring of indeterminates, and then it has meaning when paired with an evaluation homomorphism

But that's just with polynomials and of course there are many uses of x to define functions or other things that aren't polynomials

How is it done in the world of logic? I need help Doug I don't know things

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What the symbol 'x' means depends on the author using it and the context.

@DougSpoonwood yeah but what does it mean in the context when you call it a variable?

When I call 'x' a variable I mean that there exists more than one possibility that can get put in the place of 'x' wherever it appears. Within the integers, I wouldn't call 'x' (x+2)=4 a variable, since only one natural number can get put in place of 'x'. But, I would call 'x' in (xx) = 4 a variable, since more than one integer can get put in place of 'x' and (xx) = 4 hold true.

By 'xx', I mean the product of 'x' and 'x', as in integer multiplication. Using the star symbol defaulted to the coding scheme. Perhaps I should have written something more like the following: If I call 'X' a variable, I mean that there exists more than one possibility that can get put in the place of 'X' wherever it appears. Within the integers, I wouldn't call 'X' in (X+2)=4 a variable, since only one natural number can get put in place of 'X'. But, I would call 'X' in (X x X) = 4 a variable.