About the transcendental controlling the physical

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godelian
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About the transcendental controlling the physical

Post by godelian »

Let's say that we have a standard universe (of natural numbers).

According to Lowenheim-Skolem theorem, we also have nonstandard universes of nonstandard numbers. These universes are transcendental with regards to the standard one.

Now let's look at a particular pattern visible in the standard natural numbers, i.e. Goodstein's theorem:
Wikipedia on "Goodstein's theorem" wrote: In mathematical logic, Goodstein's theorem is a statement about the natural numbers, proved by Reuben Goodstein in 1944, which states that every Goodstein sequence eventually terminates at 0.
It is impossible to prove this pattern from anything in the standard universe:
Wikipedia on "Goodstein's theorem" wrote: Kirby and Paris[1] showed that it is unprovable in Peano arithmetic.
However, Goodstein's theorem is still perfectly provable. Below an informal explanation of how the proof works:
Mark Kim Mulgrew explaining the proof wrote: Admittedly, it is still not very clear how we would go about creating a strictly decreasing sequence that bounds the Goodstein sequence from above. Surely, we would need a sequence of very large numbers to do this. In fact, wouldn't a sequence of "decreasing infinities" (whatever that means) do the job? "Infinities" would of course be larger than any natural number, and then we might hope to be able to generalize the well-ordering principle to these "infinities" to conclude that the "decreasing sequence of infinities" terminates to zero.
Hence, the only way to explain this pattern visible in the standard universe is from logic action that originates in a transcendental universe.

We need transfinite ordinals in a nonstandard transcendental universe for this.

Hence, the transcendental universe controls the standard universe.

Is this an uncommon way of thinking?

No, it isn't.

In religion, it is even standard doctrine to believe that the transcendental controls the physical.
Skepdick
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Re: About the transcendental controlling the physical

Post by Skepdick »

godelian wrote: Sat Jun 25, 2022 4:24 am Let's say that we have a standard universe (of natural numbers).

According to Lowenheim-Skolem theorem, we also have nonstandard universes of nonstandard numbers. These universes are transcendental with regards to the standard one.

Now let's look at a particular pattern visible in the standard natural numbers, i.e. Goodstein's theorem:
Wikipedia on "Goodstein's theorem" wrote: In mathematical logic, Goodstein's theorem is a statement about the natural numbers, proved by Reuben Goodstein in 1944, which states that every Goodstein sequence eventually terminates at 0.
It is impossible to prove this pattern from anything in the standard universe:
Wikipedia on "Goodstein's theorem" wrote: Kirby and Paris[1] showed that it is unprovable in Peano arithmetic.
However, Goodstein's theorem is still perfectly provable. Below an informal explanation of how the proof works:
Mark Kim Mulgrew explaining the proof wrote: Admittedly, it is still not very clear how we would go about creating a strictly decreasing sequence that bounds the Goodstein sequence from above. Surely, we would need a sequence of very large numbers to do this. In fact, wouldn't a sequence of "decreasing infinities" (whatever that means) do the job? "Infinities" would of course be larger than any natural number, and then we might hope to be able to generalize the well-ordering principle to these "infinities" to conclude that the "decreasing sequence of infinities" terminates to zero.
Hence, the only way to explain this pattern visible in the standard universe is from logic action that originates in a transcendental universe.

We need transfinite ordinals in a nonstandard transcendental universe for this.

Hence, the transcendental universe controls the standard universe.

Is this an uncommon way of thinking?

No, it isn't.

In religion, it is even standard doctrine to believe that the transcendental controls the physical.
There's nothign interesting about this - it's just an elaborate tautology.

That's precisely what "provability" means. Given a set of axioms does a particular theorem hold?

If Goodstein's theorem doesn't hold in the PA axioms then it's nonsensical to speak about the Godstein "theorem" in the PA universe.
In fact, when you are talking about the Goodstein's theorem then you are necessarily constraining the domain of discourse precisely to the universes in which the theorem is provable; and you are excluding the universes in which the theorem is not provable.

SELECT * FROM multiverse WHERE Provable(Goodstein-claim)

The "transcendental" is that which chooses the axioms. The "transcendental" is you.

Outside the world of mathematics what you are observing is nothing but downward causation. The mind affecting reality. Just as reality affects the mind.
godelian
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Re: About the transcendental controlling the physical

Post by godelian »

Skepdick wrote: Sat Jun 25, 2022 10:01 am If Goodstein's theorem doesn't hold in the PA axioms then it's nonsensical to speak about the Godstein "theorem" in the PA universe.
Goodstein's theorem is true in the standard universe ("model" or "interpretation") of the natural numbers, i.e. "the intended interpretation", but not provable from PA, which is the most common theory for constructing the natural numbers.

The following math stackexchange question is actually a good discussion about what that means. In the answers, they rather refer to the Parris-Harrington theorem than the Goodstein theorem, as it is another theorem that is similarly true in the universe of natural numbers but not provable from PA.
https://math.stackexchange.com/questions/213253/what-does-it-mean-for-something-to-be-true-but-not-provable-in-peano-arithmetic

Question: What does it mean for something to be true but not provable in peano arithmetic?

Answer1:

Peano Arithmetic is a particular proof system for reasoning about the natural numbers. As such it does not make sense to speak about something being "true in PA" -- there is only "provable in PA", "disprovable in PA", and "independent of PA".

When we speak of "truth" it must be with respect to some particular model. In the case of arithmetic statements, the model we always speak about unless something else is explicitly specified is the actual (Platonic) natural numbers. Virtually all mathematicians expect these numbers to "exist" (in whichever philosophical sense you prefer mathematical objects to exist in) independently of any formal system for reasoning about them, and the great majority expect all statements about them to have objective (but not necessarily knowable) truth values.

We're very sure that everything that is "provable in PA" is also "true about the natural numbers", but the converse does not hold: There exist sentences that are "true about the actual natural numbers" but not "provable in PA". This was famously proved by Gödel -- actually he gave a (formalizable, with a few additional technical assumptions) proof that a particular sentence was neither "provable in PA" nor "disprovable in PA", and a convincing (but not strictly formalizable) argument that this sentence is true about the actual natural numbers.

Paris-Harrington shows that another particular sentence is of this kind: not provable in PA, yet true about the actual natural numbers.

Answer 2:

You seem to be confused about the meaning of true when it comes to peano arithmetic. Peano arithmetic usually refers to a particular set of axioms about the natural numbers, formalized in first-order predicate logic.

If you have a statement A and a set of axioms T:

* it can be that A is provable in T. A is then true in every model of T (A model is a particular implementation of a set of axioms if you will - it assigns actual values to all constant symbols which occur in the axioms, actual functions to all functions which occur in the axioms, and so on).

* A is true in every model of T. In first-order predicate logic this implies that A is provable in T. (Note that this is not the case in second-order predicate logic!). Some people might abbreviate this as "A is true in T", but that may be misunderstood as "A is true in the standard model of T
", so this abbreviation should be avoided.

* A is true in a particular model of T. This implies nothing about the truth of A in other models, and nothing about whether A is provable. It does imply, however, that the opposite of A cannot be provable, otherwise that particular model would have to satisfy A and its opposite.

* A contradicts T. This is the same as T proving the opposite of A, and obviously no model can then fullfil A, since all fulfil the opposite of A.

In the case of PA, people might also say "True over the integers" or simply "true" instead of "True over the standard model of PA". Especially because PA is not the only axiomatization of the integers.

The generalized Ramsey theorem (sometimes called Paris-Harrington Principle, btw) is not provable in PA, but it is true in the standard model of PA, i.e. in the usual integers. We know that because it is provable in second-order arithmetic, i.e. a second-order axiomatization of the integers.
Skepdick
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Re: About the transcendental controlling the physical

Post by Skepdick »

godelian wrote: Sat Jun 25, 2022 11:28 am Goodstein's theorem is true in the standard universe ("model" or "interpretation") of the natural numbers, i.e. "the intended interpretation", but not provable from PA, which is the most common theory for constructing the natural numbers.
If it's not provable in a model then it's not a theorem in the model.

Because "theorem" means "provable". Theorem does not mean true.
theorem /ˈθɪərəm/ noun a general proposition not self-evident but proved by a chain of reasoning; a truth established by means of accepted truths.
You are flip-flopping between your model-theoretic (semantic) and your proof-theoretic (syntactic) perspectives.
godelian
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Re: About the transcendental controlling the physical

Post by godelian »

Skepdick wrote: Sat Jun 25, 2022 11:36 am If it's not provable in a model then it's not a theorem in the model.
Because "theorem" means "provable". Theorem does not mean true.
Concerning soundness:
wikipedia on "soundness" wrote: Soundness of a deductive system is the property that any sentence that is provable in that deductive system is also true on all interpretations or structures of the semantic theory for the language upon which that theory is based. In symbols, where S is the deductive system, L the language together with its semantic theory, and P a sentence of L: if ⊢S P, then also ⊨L P.
A theorem (which is always a logic sentence) that is provable in a theory is therefore true in all its models/interpretations (theorem of soundness).
Skepdick
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Re: About the transcendental controlling the physical

Post by Skepdick »

godelian wrote: Sat Jun 25, 2022 12:03 pm
Skepdick wrote: Sat Jun 25, 2022 11:36 am If it's not provable in a model then it's not a theorem in the model.
Because "theorem" means "provable". Theorem does not mean true.
Concerning soundness:
wikipedia on "soundness" wrote: Soundness of a deductive system is the property that any sentence that is provable in that deductive system is also true on all interpretations or structures of the semantic theory for the language upon which that theory is based. In symbols, where S is the deductive system, L the language together with its semantic theory, and P a sentence of L: if ⊢S P, then also ⊨L P.
A theorem (which is always a logic sentence) that is provable in a theory is therefore true in all its models/interpretations (theorem of soundness).
Yes. Translated into our particular scenario.

If the Goodstein statement/sentence was provable in PA (e.g it was a theorem of PA) then it would also be true in all models of PA.

It's not provable in PA. So it may or may not be true in all models of PA.
godelian
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Re: About the transcendental controlling the physical

Post by godelian »

Skepdick wrote: Sat Jun 25, 2022 12:13 pm If the Goodstein statement/sentence was provable in PA (e.g it was a theorem of PA)
It may not be provable in PA but it can be expressed in the language of PA, i.e. the language used to phrase it's axioms. That is enough for a logic sentence to be "of PA".

Any logic sentence that can be expressed in the language in which the axioms of a theory have been expressed, is a logic sentence of that theory.

Such logic sentence may, or may not, be provable from that theory. In fact, that was exactly the question in Hilbert's Etscheidungsproblem:

Are all the logic sentences in the language of a theory, provable (or disprovable) from that theory?
Wikipedia on "Entscheidungsproblem" wrote: By the completeness theorem of first-order logic, a statement is universally valid if and only if it can be deduced from the axioms, so the Entscheidungsproblem can also be viewed as asking for an algorithm to decide whether a given statement is provable from the axioms using the rules of logic.

In 1936, Alonzo Church and Alan Turing published independent papers[2] showing that a general solution to the Entscheidungsproblem is impossible.
So, a logic sentence legitimately belongs to a theory, if it can be expressed in the language of that theory.

For example, the Riemann hypothesis belongs to number theory because the question can be phrased in the language of number theory. The fact that it seems to be unprovable, does not matter:

[quote Wikipedia on "Riemann hypothesis"]
In mathematics, the Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1/2. Many consider it to be the most important unsolved problem in pure mathematics.[1] It is of great interest in number theory because it implies results about the distribution of prime numbers.
[/quote]

Fermat's last theorem has always been a question in Peano Arithmetic (PA), even when there was no proof already, because it is expressed in the language of PA.
then it would also be true in all models of PA.
Skepdick wrote: Sat Jun 25, 2022 12:13 pm It's not provable in PA. So it may or may not be true in all models of PA.
Yes. It is true in the natural numbers. However, there must be nonstandard models of PA in which it is false.

Otherwise, if Goodstein's theorem were true in all models of PA, then according to Godel's semantic completeness theorem, there would have to exist a proof for it from PA. Paris and Kirby have proven, however, that this proof cannot exist in PA.
Skepdick
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Re: About the transcendental controlling the physical

Post by Skepdick »

godelian wrote: Sat Jun 25, 2022 1:04 pm It may not be provable in PA but it can be expressed in the language of PA, i.e. the language used to phrase it's axioms. That is enough for a logic sentence to be "of PA".

Any logic sentence that can be expressed in the language in which the axioms of a theory have been expressed, is a logic sentence of that theory.

Such logic sentence may, or may not, be provable from that theory. In fact, that was exactly the question in Hilbert's Etscheidungsproblem:

Are all the logic sentences in the language of a theory, provable (or disprovable) from that theory?
The point is that if you adopt the perspective of the internal language of PA the Goodstein sentence is not a theorem. And in this perspective (being all syntactic) any speak of truth is meaningless. In this perspective you are a proof-theorist.

If you want to speak about truth you need to adopt a perspective (a language) external to the theory. And in this perspective you are a model-theorist.
godelian wrote: Sat Jun 25, 2022 1:04 pm Yes. It is true in the natural numbers. However, there must be nonstandard models of PA in which it is false.

Otherwise, if Goodstein's theorem were true in all models of PA, then according to Godel's semantic completeness theorem, there would have to exist a proof for it from PA. Paris and Kirby have proven, however, that this proof cannot exist in PA.
Hence - you can't call it a theorem in PA.
godelian
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Re: About the transcendental controlling the physical

Post by godelian »

Skepdick wrote: Sat Jun 25, 2022 1:25 pm The point is that if you adopt the perspective of the internal language of PA the Goodstein sentence is not a theorem. And in this perspective (being all syntactic) any speak of truth is meaningless. In this perspective you are a proof-theorist.
It is known as a theorem because it is provable (outside PA). In PA itself, it is a logic sentence or statement.

The wording in Wikipedia:
In mathematical logic, Goodstein's theorem is a statement about the natural numbers, proved by Reuben Goodstein in 1944, which states that every Goodstein sequence eventually terminates at 0.
The wording in Wolfram:
Even more amazingly, Paris and Kirby showed in 1982 that Goodstein's theorem is not provable in ordinary Peano arithmetic (Borwein and Bailey 2003, p. 35).
Below a paper that classifies nonstandard models of arithmetic according to whether Goodstein's theorem is true in it, or not:
A CLASSIFICATION OF NONSTANDARD MODELS OF PEANO ARITHMETIC BY GOODSTEIN'S THEOREM

DAN KAPLAN
Abstract. In this paper I intend to outline a method for finding nonstandard models of Peano Arithmetic ( PA ) that satisfy Goodstein's theorem. Goodstein's Theorem is an interesting result because, though it is expressible completely in the language of number theory, it is nonetheless independent of the axioms of PA . I begin by rehearsing a proof of Goodstein's theorem, followed by a proof of its independence, developing the necessary tools to do so along the way. Finally, using indicator theory, I show how that can classify the nonstandard models according to Goodstein's Theorem.
Skepdick wrote: Sat Jun 25, 2022 1:25 pm If you want to speak about truth you need to adopt a perspective (a language) external to the theory.
Kaplan expresses it as: "it is expressible completely in the language of number theory, it is nonetheless independent of the axioms of PA".

The language in which Goodstein's logic sentence is expressed, is the one of PA, which is also the same as the language of the model (the natural numbers). There is no difference in language between theory and model. The only thing that the model does, is to bind the non-logical symbols, i.e. some part of the vocabulary in the language, to one particular interpretation.
Skepdick wrote: Sat Jun 25, 2022 1:25 pm Hence - you can't call it a theorem in PA.
"Goodstein's theorem" is known as a "theorem" because it is provable from some theory (other than PA). In PA itself, it is a logic sentence (or a statement) which its standard model, the natural numbers, happens to satisfy.
Phil8659
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Re: About the transcendental controlling the physical

Post by Phil8659 »

Once upon a time, I had a transcendental possum. My friends just called it road kill.

Now, answer me this, since a number is no more than a method of naming in the arithmetic system of grammar and is only one of the four ways in which binary recursion can be effected by a grammar system, is it not the most elementary form of stupidity to claim that what we do, make standards of names to effect binary recursion, exampled by the very definition of a thing itself as a relative contained by correlatives, to call our very actions, a theory? So, are we then a theory of a theory of a theory?
If you cannot dazzled them with brilliance then baffle them with bull shit, a whole lot of pseudo-intellectuals just like the smell.

Is their any way, that you can demonstrate that a system of grammar is a theory of itself? Duh!.
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