Tarski Undefinability Theorem Succinctly Refuted

What is the basis for reason? And mathematics?

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wtf
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Re: Tarski Undefinability Theorem Reexamined

Post by wtf » Thu Apr 04, 2019 9:34 pm

PeteOlcott wrote:
Thu Apr 04, 2019 4:02 am
It turns out that axioms are the ultimate foundation of Truth
I'm fascinated as to how anyone could hold such an opinion.

Supposes we are in the realm of math, and we consider the axiom of choice (AC). It's well-known that AC is independent of the other axioms of ZF set theory.

So if we do math with AC, then AC is an axiom and we derive consequences.

And if we do math with the negation of AC, then not-AC is an axiom and we derive consequences.

In what way can you say AC is therefore both true and false at the same time? If one is a Platonist, one believes that there is an ultimate truth of the matter, which ZF simply fails to capture. And if one is a formalist, one understands that AC is neither true or false, in the same way that the rules of chess are neither true nor false but rather simply arbitrary rules in a formal game.

So WTF are you talking about?

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A_Seagull
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Re: Tarski Undefinability Theorem Succinctly Refuted

Post by A_Seagull » Thu Apr 04, 2019 9:47 pm

Speakpigeon wrote:
Thu Apr 04, 2019 5:46 pm
PeteOlcott 1 - A_Seagull 0
Popcorn anyone?
EB
Lol

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A_Seagull
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Re: Tarski Undefinability Theorem Reexamined

Post by A_Seagull » Thu Apr 04, 2019 9:50 pm

PeteOlcott wrote:
Thu Apr 04, 2019 4:06 pm
A_Seagull wrote:
Thu Apr 04, 2019 10:28 am
PeteOlcott wrote:
Thu Apr 04, 2019 4:47 am


If the English language actually defines nothing then you didn't just say that in English.
Lol
Unless you are just playing games I would estimate that you may not have a deep enough
understanding of these things to provide any useful feedback.
Thank you, but I prefer not to debate with children.

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Re: Tarski Undefinability Theorem Reexamined

Post by PeteOlcott » Thu Apr 04, 2019 11:56 pm

wtf wrote:
Thu Apr 04, 2019 9:34 pm
PeteOlcott wrote:
Thu Apr 04, 2019 4:02 am
It turns out that axioms are the ultimate foundation of Truth
I'm fascinated as to how anyone could hold such an opinion.

Supposes we are in the realm of math, and we consider the axiom of choice (AC). It's well-known that AC is independent of the other axioms of ZF set theory.

So if we do math with AC, then AC is an axiom and we derive consequences.

And if we do math with the negation of AC, then not-AC is an axiom and we derive consequences.

In what way can you say AC is therefore both true and false at the same time? If one is a Platonist, one believes that there is an ultimate truth of the matter, which ZF simply fails to capture. And if one is a formalist, one understands that AC is neither true or false, in the same way that the rules of chess are neither true nor false but rather simply arbitrary rules in a formal game.

So WTF are you talking about?
There is a crucial key detail that you are leaving out: What consequences are AC and ~AC deriving?
There is no contradiction when an axiom and its negation derive consequences.
There is only a contradiction when an axiom and its negation derive THE SAME consequence.

wtf
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Re: Tarski Undefinability Theorem Reexamined

Post by wtf » Fri Apr 05, 2019 12:26 am

PeteOlcott wrote:
Thu Apr 04, 2019 11:56 pm
There is a crucial key detail that you are leaving out: What consequences are AC and ~AC deriving?
There is no contradiction when an axiom and its negation derive consequences.
There is only a contradiction when an axiom and its negation derive THE SAME consequence.
That's not even a coherent response to the question I put to you. Nor do you have the slightest clue what a contradiction is. Back to logic 101 for you.

PeteOlcott
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Re: Tarski Undefinability Theorem Reexamined

Post by PeteOlcott » Fri Apr 05, 2019 1:11 am

wtf wrote:
Thu Apr 04, 2019 9:34 pm
So if we do math with AC, then AC is an axiom and we derive consequences.

And if we do math with the negation of AC, then not-AC is an axiom and we derive consequences.

In what way can you say AC is therefore both true and false at the same time?
So WTF are you talking about?
Ah, I see that my prior reply was incorrect.

There are no expressions of language that are both True and False.
There are plenty that are neither True nor False.

Some semantically ill-formed expressions of language are neither True nor False.

Self-contradictory expressions of language are treated like deductive inference
on the basis of contradictory premises thus unsound thus and not true.

wtf
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Re: Tarski Undefinability Theorem Reexamined

Post by wtf » Fri Apr 05, 2019 7:18 am

PeteOlcott wrote:
Fri Apr 05, 2019 1:11 am

Ah, I see that my prior reply was incorrect.
This one also does not address my question. You be dancin' when you should be clear and direct.

Originally you said:
PeteOlcott wrote:
Thu Apr 04, 2019 4:02 am
It turns out that axioms are the ultimate foundation of Truth
So I put to you this situation. There are two mathematicians. Following the custom of cryptographers we will call them Alice and Bob. Alice is pro-choice (bad pun ok). She accepts the axiom of choice. In her world you can pick an element from each of a collection of nonempty sets. Every vector space has a basis. A set is infinite if and only if it's Dedekind-infinite. The Banach-Tarski paradox is a theorem.

Bob, on the other hand, is no-choice. He rejects the axiom of choice. In his world, there's a collection of nonempty sets without a choice function. There's a vector space without a basis. There's an infinite set that's not Dedekind-infinite. The Banach-Tarski paradox is NOT a theorem.

Now you say that axioms determine truth. So what is the truth of the axiom of choice?

Now to be fair I used an example only familiar to people who have studied some math. I used the axiom of choice because it's abstract and it's clear that it has a definite truth value only to a Platonist.

But I could just as easily have put to you the same question about the parallel postulate. We have a consistent geometry in which there's exactly one parallel to a line through a given point not on the line. We have consistent geometries where there are no such parallels, and consistent geometries where there are many such parallels.

Which geometry is true? It's a question of physics; so unlike with the axiom of choice, there is a definite truth of the matter. However, the axioms of geometry do not resolve the issue. To determine the truth of the parallel postulate, one must examine physical reality using scientific experiment. This is an active research area of cosmology. Real life physicists and astronomers go to work each day and attempt to discover the true geometry of the universe. But they cannot rely on axioms. They must make physical observations.

These two examples, one abstract and the other physical, falsify your claim that axioms determine Truth (your capitalization).

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Re: Tarski Undefinability Theorem Reexamined

Post by PeteOlcott » Fri Apr 05, 2019 3:53 pm

wtf wrote:
Fri Apr 05, 2019 7:18 am
PeteOlcott wrote:
Fri Apr 05, 2019 1:11 am

Ah, I see that my prior reply was incorrect.
This one also does not address my question. You be dancin' when you should be clear and direct.

Originally you said:
PeteOlcott wrote:
Thu Apr 04, 2019 4:02 am
It turns out that axioms are the ultimate foundation of Truth
So I put to you this situation. There are two mathematicians. Following the custom of cryptographers we will call them Alice and Bob. Alice is pro-choice (bad pun ok). She accepts the axiom of choice. In her world you can pick an element from each of a collection of nonempty sets. Every vector space has a basis. A set is infinite if and only if it's Dedekind-infinite. The Banach-Tarski paradox is a theorem.

Bob, on the other hand, is no-choice. He rejects the axiom of choice. In his world, there's a collection of nonempty sets without a choice function. There's a vector space without a basis. There's an infinite set that's not Dedekind-infinite. The Banach-Tarski paradox is NOT a theorem.

Now you say that axioms determine truth. So what is the truth of the axiom of choice?

Now to be fair I used an example only familiar to people who have studied some math. I used the axiom of choice because it's abstract and it's clear that it has a definite truth value only to a Platonist.

But I could just as easily have put to you the same question about the parallel postulate. We have a consistent geometry in which there's exactly one parallel to a line through a given point not on the line. We have consistent geometries where there are no such parallels, and consistent geometries where there are many such parallels.

Which geometry is true? It's a question of physics; so unlike with the axiom of choice, there is a definite truth of the matter. However, the axioms of geometry do not resolve the issue. To determine the truth of the parallel postulate, one must examine physical reality using scientific experiment. This is an active research area of cosmology. Real life physicists and astronomers go to work each day and attempt to discover the true geometry of the universe. But they cannot rely on axioms. They must make physical observations.

These two examples, one abstract and the other physical, falsify your claim that axioms determine Truth (your capitalization).
Within imaginary number theory the square root of -1 is an axiom.
Within the universal set of all knowledge this axiom is rejected on the basis of contradicting other axioms.

wtf
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Re: Tarski Undefinability Theorem Reexamined

Post by wtf » Fri Apr 05, 2019 4:53 pm

PeteOlcott wrote:
Fri Apr 05, 2019 3:53 pm
Within imaginary number theory the square root of -1 is an axiom.
Within the universal set of all knowledge this axiom is rejected on the basis of contradicting other axioms.
Are you utterly incapable of engaging with a substantive point? You claimed axioms determine truth. I gave you two solid counterexamples. You've now avoided responding for three consecutive posts. Why bother responding at all then?

PeteOlcott
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Re: Tarski Undefinability Theorem Reexamined

Post by PeteOlcott » Fri Apr 05, 2019 5:55 pm

wtf wrote:
Fri Apr 05, 2019 4:53 pm
PeteOlcott wrote:
Fri Apr 05, 2019 3:53 pm
Within imaginary number theory the square root of -1 is an axiom.
Within the universal set of all knowledge this axiom is rejected on the basis of contradicting other axioms.
Are you utterly incapable of engaging with a substantive point? You claimed axioms determine truth. I gave you two solid counterexamples. You've now avoided responding for three consecutive posts. Why bother responding at all then?
I have not paid hardly any attention to ZFC or anything related to ZFC
such as the Axiom of choice.

To eliminate Russell's paradox I would have simply added this axiom to
naive set theory: ∀x (x ∉ x)

This axiom would be inherited from its BASE class in the following knowledge ontology:

∀x ∉ Thing (~Totally_Contains(x, x))
---∀x ∉ Physical_Thing (~Totally_Contains(x, x))
---∀x ∉ Conceptual_Thing (~Totally_Contains(x, x))
------∀x ∉ Set (~Totally_Contains(x, x))

wtf
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Re: Tarski Undefinability Theorem Reexamined

Post by wtf » Fri Apr 05, 2019 6:36 pm

PeteOlcott wrote:
Fri Apr 05, 2019 5:55 pm
wtf wrote:
Fri Apr 05, 2019 4:53 pm
PeteOlcott wrote:
Fri Apr 05, 2019 3:53 pm
Within imaginary number theory the square root of -1 is an axiom.
Within the universal set of all knowledge this axiom is rejected on the basis of contradicting other axioms.
Are you utterly incapable of engaging with a substantive point? You claimed axioms determine truth. I gave you two solid counterexamples. You've now avoided responding for three consecutive posts. Why bother responding at all then?
I have not paid hardly any attention to ZFC or anything related to ZFC
such as the Axiom of choice.

To eliminate Russell's paradox I would have simply added this axiom to
naive set theory: ∀x (x ∉ x)

This axiom would be inherited from its BASE class in the following knowledge ontology:

∀x ∉ Thing (~Totally_Contains(x, x))
---∀x ∉ Physical_Thing (~Totally_Contains(x, x))
---∀x ∉ Conceptual_Thing (~Totally_Contains(x, x))
------∀x ∉ Set (~Totally_Contains(x, x))
In other words: Yes. You are utterly incapable of engaging with my questions.

Your claim that axioms have anything at all to do with truth stands refuted. As it has been since the discovery of non-Euclidean geometry in the 1840's.
PeteOlcott wrote:
Fri Apr 05, 2019 5:55 pm
To eliminate Russell's paradox I would have simply added this axiom to
naive set theory: ∀x (x ∉ x)
And what if x ∈ y ∈ z ∈ x?

This problem has already been solved in set theory

https://en.wikipedia.org/wiki/Axiom_of_regularity

But what has Russell's paradox to do with ANYTHING I asked you?

You claimed (repeatedly on this website) that axioms determine truth. Of course that is laughably wrong, since axioms are syntactic and truth is semantic. I posed two counterexamples, which you haven't been able to engage with or even acknowledge.

PeteOlcott
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Re: Tarski Undefinability Theorem Reexamined

Post by PeteOlcott » Fri Apr 05, 2019 7:09 pm

wtf wrote:
Fri Apr 05, 2019 6:36 pm
PeteOlcott wrote:
Fri Apr 05, 2019 5:55 pm
wtf wrote:
Fri Apr 05, 2019 4:53 pm


Are you utterly incapable of engaging with a substantive point? You claimed axioms determine truth. I gave you two solid counterexamples. You've now avoided responding for three consecutive posts. Why bother responding at all then?
I have not paid hardly any attention to ZFC or anything related to ZFC
such as the Axiom of choice.

To eliminate Russell's paradox I would have simply added this axiom to
naive set theory: ∀x (x ∉ x)

This axiom would be inherited from its BASE class in the following knowledge ontology:

∀x ∉ Thing (~Totally_Contains(x, x))
---∀x ∉ Physical_Thing (~Totally_Contains(x, x))
---∀x ∉ Conceptual_Thing (~Totally_Contains(x, x))
------∀x ∉ Set (~Totally_Contains(x, x))
In other words: Yes. You are utterly incapable of engaging with my questions.

Your claim that axioms have anything at all to do with truth stands refuted. As it has been since the discovery of non-Euclidean geometry in the 1840's.
PeteOlcott wrote:
Fri Apr 05, 2019 5:55 pm
To eliminate Russell's paradox I would have simply added this axiom to
naive set theory: ∀x (x ∉ x)
And what if x ∈ y ∈ z ∈ x?

This problem has already been solved in set theory

https://en.wikipedia.org/wiki/Axiom_of_regularity

But what has Russell's paradox to do with ANYTHING I asked you?

You claimed (repeatedly on this website) that axioms determine truth. Of course that is laughably wrong, since axioms are syntactic and truth is semantic. I posed two counterexamples, which you haven't been able to engage with or even acknowledge.
The ONLY reason that anyone knows that a {dog} is not a type of {dump truck}
is that we have expressions of language that have had their Boolean property assigned
the value of true.

You keep asking me about the Axiom of Choice and I have no idea what that is.

You say that syntax and semantics are necessarily two different things
even after I give you an example of the syntax of semantics:
http://liarparadox.org/Meaning_Postulat ... p_1952.pdf

wtf
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Re: Tarski Undefinability Theorem Reexamined

Post by wtf » Fri Apr 05, 2019 7:14 pm

PeteOlcott wrote:
Fri Apr 05, 2019 7:09 pm
You keep asking me about the Axiom of Choice and I have no idea what that is.
Yet another deflection, since I also asked you about non-Euclidean geometry, which is a question of physics clearly NOT resolved by any axioms.
PeteOlcott wrote:
Fri Apr 05, 2019 7:09 pm
You say that syntax and semantics are necessarily two different things
Uh ... yeah. That's because they are two different things. Syntax is symbol manipulation. Semantics is meaning.
PeteOlcott wrote:
Fri Apr 05, 2019 7:09 pm
even after I give you an example of the syntax of semantics:
http://liarparadox.org/Meaning_Postulat ... p_1952.pdf
Another deflection. But ‎Gödel was obsessively interested in set theory. He proved that the axiom of choice was consistent with ZF. For you to claim to have falsified his incompleteness theorems while being utterly unaware of his work shows the shallowness of your own thinking and study.

I'm done here. You have no idea what you're talking about and can't defend your own mistaken ideas.

Logik
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Re: Tarski Undefinability Theorem Reexamined

Post by Logik » Fri Apr 05, 2019 7:36 pm

wtf wrote:
Fri Apr 05, 2019 7:14 pm
Syntax is symbol manipulation. Semantics is meaning.
Interjection (or a rude derailment). Part of solving the symbol-grounding problem ( https://en.wikipedia.org/wiki/Symbol_grounding_problem ) is the unification of syntax and semantics.

And so you get to ask the question: Has the symbol-grounding problem been solved?

My meaning is deterministic when interpreted by a physical machine called a computer. My words (bashing on a keyboard) have physical, real-world consequences (mutating memory on computers, altering voltages on routers transmitting packets etc. etc.).

I am intentionally pushing the buttons on my keyboard knowing that you are going to be reading this text very shortly.

Bar for the fact that you may (mis?)interpret what I am saying I would be willing to argue for a position that computers ground symbols.

PeteOlcott
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Re: Tarski Undefinability Theorem Reexamined

Post by PeteOlcott » Fri Apr 05, 2019 7:58 pm

wtf wrote:
Fri Apr 05, 2019 7:14 pm
PeteOlcott wrote:
Fri Apr 05, 2019 7:09 pm
even after I give you an example of the syntax of semantics:
http://liarparadox.org/Meaning_Postulat ... p_1952.pdf
Another deflection.
Not a deflection a necessary prerequisite to further communication.
Unless and and until you fully understand how semantics is expressed using syntax you don't know enough to critique me.

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