Cont: Deeper than primes - Continuation 2

[doronshadmi]

This still doesn't make sense. First set A was a set, then a "formal system", now it's an extension. And it contains things that it already contained? And it must be taken?

From my first post about A, A is a formal system of infinitely many wffs (which is strong enough in order to deal with Arithmetic, exactly because it is an extension of ZF(C)), where all the infinitely many wffs are already included in A, exactly because Infinity is taken in terms of Platonic (or Actual) Infinity (By Platonic (or Actual) Infinity there exists a set of infinitely many things (for example: wffs) as a complete whole).

The maneuvers of jsfisher around A's existence, this is exactly the thing that makes no sense.

Here is jsfisher's last reply, which clearly demonstrates his nonsensical maneuvers around A's existence:

That's nice, but you still haven't told us how you are actually applying the axiom on ZF(C). Keep in mind, the axiom simply states that there exists a certain set with certain properties. Nothing more.

So, once again, what does "using ZF(C) Axiom Of Infinity on ZF(C) itself" mean?

and he does them in order to avoid the following question:

Please explain what do you mean by "The Axiom of Infinity establishes a set in terms of Mathematics." (especially the highlighted part)?

Keep in mind, the axiom simply states that there exists a certain set with certain properties. Nothing more.

Keep in mind that ZF(C) Axiom Of Infinity is taken in terms of Platonic (or Actual) Infinity, such that there a exists a certain set with infinitely many things as a complete whole.

I take the property of Platonic Infinity from ZF(C) Axiom Of Infinity and relate it to A. Nothing more.

Now, please explain what do you mean by "The Axiom of Infinity establishes a set in terms of Mathematics." (especially the highlighted part)?

(To the other posters:

"In the philosophy of mathematics, the abstraction of actual infinity involves the acceptance (if the axiom of infinity is included) of infinite entities, such as the set of all natural numbers or an infinite sequence of rational numbers, as given, actual, completed objects."

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

Moreover

"Historically, logic has been studied in philosophy (since ancient times) and mathematics (since the mid-19th century), and recently logic has been studied in cognitive science (encompasses computer science, linguistics, philosophy and psychology"

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

Also, Infinity is one of the main philosophical subjects, studied by philosophers like Plato, Aristotle and many more philosophers along the years.

The attepmt to define a clear cut distinction between Philosophy and Mathematics in case of Logic and Infinity, is itself some kind of Philosophy, and in this case jsfisher's philosophy about the discussed subject)

 

[jsfisher]

Keep in mind that ZF(C) Axiom Of Infinity is taken in terms of Platonic (or Actual) Infinity, such that there a exists a certain set with infinitely many things as a complete whole.

Almost. The Axiom is not "taken"; it simply is. A certain set exists; it has certain properties. The von Neumann ordinal is the minimal example of such a set, so we have one example of the set guaranteed to exist. The Axiom alone gives no guidance as to whether there are others.

Your insistence on bringing in philosophic babble is, well, yours.

I take the property of Platonic Infinity from ZF(C) Axiom Of Infinity and relate it to A. Nothing more.

Great, you've gone full circular on us. To get A, you "take the property of Platonic Infinity from ZF(C) Axiom Of Infinity and relate it to A."

You probably want to clean that up, and when you do, please explain what "relate it to" means.

 

[doronshadmi]

Almost. The Axiom is not "taken"; it simply is.

Let's see: "The Axiom is ... simply is."

jsfisher, maybe this is a very interesting statement. Probably a lot of mathematical work can by done by it and maybe also a profound communication between people can be done by it.

Unfortunately, I do not find such tautology as very useful in our discussion.

The von Neumann ordinal is the minimal example of such a set, so we have one example of the set guaranteed to exist.

Such set is guaranteed to exist in terms of Platonic (or Actual Infinity) Infinity (which according to it there exists an infinite set as a complete whole).

Your insistence on bringing in philosophic babble is, well, yours.

Your insistence to establish a clear cut border between Philosophy and Mathematics (and in the discussed case, Logic) is, well, your philosophy.

please explain what "relate it to" means.

It means that a certain property of x is also a property of y, and in the considered case Infinity is established in terms of Platonic Infinity, both on x and y.

Please explain what do you mean by "The Axiom of Infinity establishes a set in terms of Mathematics." (especially the highlighted part)?

 

[jsfisher]

It means that a certain property of x is also a property of y, and in the considered case Infinity is established in terms of Platonic Infinity.

How would that apply to the case at hand, that being "using ZF(C) Axiom Of Infinity on ZF(C) itself" to get A. What is x; what is y; and what is this certain property?

 

[doronshadmi]

How would that apply to the case at hand, that being "using ZF(C) Axiom Of Infinity on ZF(C) itself" to get A. What is x; what is y; and what is this certain property?

The certain property is Platonic (or Actual) Infinity, which is related to ZF(C) Axiom Of Infinity (x) and ZF(C) extension .

After all natural numbers comes the first infinite ordinal, ω.

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

ω is not established without the existence of the infinite set of all natural numbers, and the infinite set of all natural numbers is not established (by ZF(C) Axiom Of Infinity) as a complete whole (which enables ω to exist "After all natural numbers") without the "philosophic babble" of Platonic (or Actual) Infinity.

Please explain what do you mean by "The Axiom of Infinity establishes a set in terms of Mathematics." (especially the highlighted part)?

 

[jsfisher]

The certain property is Platonic (or Actual) Infinity

The property would be "infinite" not "infinity", but let's move on.

which is related to ZF(C) Axiom Of Infinity (x)

How? The Axiom of Infinity is quite finite.

and ZF(C) extension .

ZF and ZFC are set theories. What is it you see that is infinite about them? And what is this extension of which you speak?

And assuming there is something infinite about the Axiom and the set theories, how does this relation of a common property between the two give rise to this set, A?


(By the way, for Z' to be an extension of Z where Z' and Z are formal systems like, say, ZF, everything that is decidable in Z must be equally decidable in Z'. There may be additional things decidable in Z', but Z' may not contradict Z.)

 

[doronshadmi]

The property would be "infinite" not "infinity", but let's move on.

Let's not move on. The certain property is Infinity in terms of Platonic (or Actual) Infinity.

How? The Axiom of Infinity is quite finite.

Platonic Infinity is what The Axiom of Infinity establishes, by using finitely many symbols, for example: The infinite set of von Neumann ordinals (https://en.wikipedia.org/wiki/Natural_number#Von_Neumann_ordinals).

ZF and ZFC are set theories. What is it you see that is infinite about them?

Axiom schema (and therefore Infinity) are parts of ZF(C).

And what is this extension of which you speak?

As done by Godel First Incompleteness Theorem, but in terms of Platonic (or Actual) Infinity (which means that all wffs (whether they are axioms or theorems) are already included in this extension (called formal system A).

And assuming there is something infinite about the Axiom and the set theories, how does this relation of a common property between the two give rise to this set, A?

Formal system A has infinitely many wffs in terms of Platonic (or Actual) Infinity (formal system A is taken as a complete whole).

(By the way, for Z' to be an extension of Z where Z' and Z are formal systems like, say, ZF, everything that is decidable in Z must be equally decidable in Z'. There may be additional things decidable in Z', but Z' may not contradict Z.)

In the considered case Z' (which is a complete extension of Z in terms of Platonic (or Actual) Infinity) is indeed strong enough to deal with Arithmetic (as decidable in Z), but unlike Z, all its infinitely many wffs are already included in it (Z' is complete in terms of Platonic Infinity) as follows:

Each wff is encoded by a Gödel number, where at least one of these wffs, called G, states "There is no number m such that m is the Gödel number of a proof in Z', of G" (since G needs a proof, it is not an axiom but a theorem).

Since all wffs are already in Z' and all Gödel numbers are already in Z' (because Infinity is taken in terms of Platonic Infinity) there is a Gödel number of a proof of G in Z', which contradicts G in Z', exactly because Z' is complete (in terms of Platonic Infinity) and therefore inconsistent exactly because Infinity is taken in terms of Platonic (or Actual) Infinity.

Conclusion: Platonic (or Actual) Infinity is the cause of the contradiction (and therefore the inconstancy) of Z' (which is an extension of Z).

---------------------

Please explain what do you mean by "The Axiom of Infinity establishes a set in terms of Mathematics." (especially the highlighted part)?

 

[jsfisher]

...snip...

Many bits of nonsense saved for a later time so as to not further defocus the current thread arc.

And what is this extension of which you speak?

As done by Godel First Incompleteness Theorem, but in terms of Platonic (or Actual) Infinity (which means that all wffs (whether they are axioms or theorems) are already included in this extension (called formal system A).

The proof of Godel's First Incompleteness Theorem is constructive in that it provides one example of an undecidable statement for an incomplete formal system. One. Not infinitely many. Just one.

And assuming there is something infinite about the Axiom and the set theories, how does this relation of a common property between the two give rise to this set, A?

Formal system A has infinitely many wffs in terms of Platonic (or Actual) Infinity (formal system A is taken as a complete whole).

Which infinitely many statements would that be? ZF (or ZFC) can be expressed as a countably infinite number of axioms; Godel can be relied upon for any finite number of additional statements. Which statements did you have mind for your set theory extension?

(By the way, for Z' to be an extension of Z where Z' and Z are formal systems like, say, ZF, everything that is decidable in Z must be equally decidable in Z'. There may be additional things decidable in Z', but Z' may not contradict Z.)

In the considered case Z' (which is a complete extension of Z in terms of Platonic (or Actual) Infinity)...

No such extension exists.

For it to exist, you would need a membership function with respect to your set, A, for the Axiom Schema of Restricted Comprehension. You just need a function that determines whether x is a member of A. You don't have one.

Merely speculating that a set must exist because you want it to does not make it so. Throwing your philosophic baggage at the problem doesn't change that.

 

[doronshadmi]

Many bits of nonsense saved for a later time so as to not further defocus the current thread arc.

Another hands waving of yours.

The proof of Godel's First Incompleteness Theorem is constructive in that it provides one example of an undecidable statement for an incomplete formal system. One. Not infinitely many. Just one.

I agree with you, G is just one statement, as written at the end of my previous post (" Z' " is used instead of "A").

Which infinitely many statements would that be? ZF (or ZFC) can be expressed as a countably infinite number of axioms;

In order to claim that there are countably infinite number of axioms, you first have to accept that there is an infinite set in terms of a complete whole.

Godel can be relied upon for any finite number of additional statements.

This is a finite GIT version that can't deduce anything about a set in terms of Platonic (or Actual) Infinity.

By GIT infinite version (in terms of Platonic (or Actual) Infinity, which, as can be seen, was not deduced by you) all the Godel numbers that encode wffs, are already in A, where one of them encodes G wff statement (which is actually a theorem, since it is proven in A).

No such extension exists.

For it to exist, you would need a membership function with respect to your set, A, for the Axiom Schema of Restricted Comprehension. You just need a function that determines whether x is a member of A. You don't have one.

The Axiom Schema of Restricted Comprehension does not exist, if Infinity is not taken in terms of Platonic (or Actual) Infinity.

If you reject what I wrote about this axiom, you also reject the existence of the infinite set of all natural numbers as a complete whole.

Throwing your philosophic baggage at the problem doesn't change that.

Your clear cut separation between Philosophy and Mathematics, does not change the fact that it is actually your Philosophy.

---------------------

Please explain what do you mean by "The Axiom of Infinity establishes a set in terms of Mathematics." (especially the highlighted part)?

 

[jsfisher]

This is a finite GIT version... .
By GIT infinite version....

You are making up your own theorems, now, too?

...all the Godel numbers that encode wffs, are already in A

If you are trying to tell us that your set, A, contains every possible statement that can be expressed in some formal language sufficient to express the axioms of ZF (or ZFC), then you have a different problem.

The set theory corresponding to your set, A, is not an extension of ZF (or ZFC). There are statements in ZF that would be contradicted in your so-called extended set theory.

As I said before, the set you claim exists does not.

You require it represent an extension to ZF that is Godel complete. No such set exists.

 

[doronshadmi]

You are making up your own theorems, now, too?

It is made up exactly as ZF(C) Axiom Of Infinity made up things in terms of Platonic (or Actual Infinity), which enables mathematicians like you to declare that the infinite set if all natural numbers, exists.

If you are trying to tell us that your set, A, contains every possible statement that can be expressed in some formal language sufficient to express the axioms of ZF (or ZFC), then you have a different problem.

The set theory corresponding to your set, A, is not an extension of ZF (or ZFC). There are statements in ZF that would be contradicted in your so-called extended set theory.

This is my argument right from the beginning of the last discussion, which is:

The very notion of Platonic (or Actual) Infinity necessarily involved with logical contradiction and therefore inconsistency, exactly because a collection of infinitely many things is taken in terms of a complete whole.

As I said before, the set you claim exists does not.

You require it represent an extension to ZF that is Godel complete. No such set exists.

Until this very moment you are still missing my argument, which is (again, since you are still missing it):

The very notion of Platonic (or Actual) Infinity necessarily involved with logical contradiction and therefore inconsistency, exactly because a collection of infinitely many things is taken in terms of a complete whole.

Because of this logical fallacy also the infinite set of all natural numbers does not exist.

Actually, the very notion of Transfinite System does not exist (in terms of logical consistency), exactly because it is established on the notion of collection of infinitely many things in terms of a complete whole.

---------------------

Again, there is a non-interesting solution about the discussed subject, as follows:

G states: "There is no number m such that m is the Godel number of a proof in A, of G"

If G is already an axiom in A (where A is an infinite set of axioms, such that Infinity is taken in terms of Platonic Infinity) it is actually a wff that is true in A, which does not have any Godel number that is used in order to encode G's proof (since axioms are true wffs that do not need any proof in A).

But then no proof is needed and mathematicians are out of job (therefore it is an unwanted solution).

---------------------

Please explain what do you mean by "The Axiom of Infinity establishes a set in terms of Mathematics." (especially the highlighted part)?

 

[jsfisher]

It is made up....

Have fun with that. Meanwhile, Mathematics not simply made up by you continues unabated by your confusion.



Oh, and the Axiom of Infinity:

...just mathematics.

 

[doronshadmi]

Have fun with that. Meanwhile, Mathematics not simply made up by you continues unabated by your confusion.



Oh, and the Axiom of Infinity:

[qimg]https://wikimedia.org/api/rest_v1/media/math/render/svg/e2d866a2b812cbd6f5e1e1709ee1585b2269bb83[/qimg] [was replaced by me according to what is written in Wikipedia]

...just mathematics.

Let's see:

"In words, there is a set I (the set which is postulated to be infinite), such that the empty set is in I, and such that whenever any x is a member of I, the set formed by taking the union of x with its singleton {x} is also a member of I." (Please compare it to https://en.wikipedia.org/wiki/Axiom_of_infinity#Formal_statement, where m is replaced by x).

If Infinity in this axiom is not taken in terms of Platonic (or Actual) Infinity, even the infinite set of all natural numbers does not exist and jsfisher's "...just mathematics" actually does not establish the Transfinite system.

---------------------

jsfisher, please explain what do you mean by "The Axiom of Infinity establishes a set in terms of Mathematics." (especially the highlighted part)?

 

[doronshadmi]

You just need a function that determines whether x is a member of A. You don't have one.

Syntactically (by formalism without semantics) There is a set A (the set which is postulated to be infinite), such that the empty set is in A, and such that whenever any x is a member of A, the set formed by taking the union of x with its singleton {x} is also a member of A.

So Syntactically x ---> xU{x} is the bijective membership function of A.

Now we are using also Semantics (adding some meaning) by establish some models about this function, as follows:


Model 1:

Let x be an axiom (wff that is not proven) in A.

Let xU{x} be a theorem (wff that is proven) in A.

Let A be an infinite set of wffs, where Infinity is taken in terms of Platonic (or Actual) Infinity (A is taken as a complete whole).

Each wff (wff that is proven) is encoded by a Gödel number, where one of these wffs, called G, states "There is no number m such that m is the Gödel number of a proof in A, of G" (since G needs a proof, it is not an axiom but a theorem).

Since all wffs (wffs that are proven) are already in A and all Gödel numbers are already in A (because Infinity is taken in terms of Platonic Infinity) there is a Gödel number of a proof of G in A, which contradicts G in A, exactly because A is complete (in terms of Platonic Infinity) and therefore inconsistent.


Model 2:

Let x or xU{x} be axioms (wff that is not proven) in A.

G axiom states: "There is no number m such that m is the Godel number of a proof in A, of G"

Since G is already an axiom in A (where A is an infinite set of axioms, such that Infinity is taken in terms of Platonic Infinity) it is actually a wff that is true in A, which does not have any Godel number that is used in order to encode G's proof (since axioms are true wffs that do not need any proof in A).

But then no proof is needed and mathematicians are out of job (therefore it is an unwanted solution).

 

[doronshadmi]

So in both models infinitely in terms of Platonic (or Actual) Infinity (which according to it there exists a collection of things as a complete whole) does not establish an interesting formal system.

An alternative to such non-interesting formal systems, is established such that Platonic (or Actual) Infinity is non-composed (it is not established in terms of collections that are taken as a complete whole) as already given in the following posts:

http://www.internationalskeptics.com/forums/showpost.php?p=12653709&postcount=3302

http://www.internationalskeptics.com/forums/showpost.php?p=12654588&postcount=3303

http://www.internationalskeptics.com/forums/showpost.php?p=12664342&postcount=3304

http://www.internationalskeptics.com/forums/showpost.php?p=12671520&postcount=3305

 

[doronshadmi]

A clearer version of my argument (as firstly was given in http://www.internationalskeptics.com/forums/showpost.php?p=12796185&postcount=3414)

Gödel numbers are used to encode wffs of formal systems that are strong enough in order to deal with Arithmetic.

In my argument, Gödel numbers are used to encode wffs as follows:

Syntactically (by formalism without semantics) there is set A (the set which is postulated to be infinite), such that the empty set is a member of A, and such that whenever any x is a member of A, the set formed by taking the union of x with its singleton {x}, is also a member of A.

So Syntactically x → xU{x} is the bijective membership function of A.

Now we are using also Semantics (adding some meaning) by establish some models about this function, as follows:


Model 1:

Let any x be an axiom (wff that is not proven) in A.

Let any xU{x} be a theorem (wff that is proven) in A.

Let A be an infinite set of wffs, where Infinity is taken in terms of Actual Infinity (A is taken as a complete whole).

Each wff (wff that is proven (some xU{x})) is encoded by a Gödel number, where one of these wffs, called G, states: "There is no number m such that m is the Gödel number of a proof in A, of G".

Since all wffs are already in A and therefore all Gödel numbers are already in A (because Infinity is taken in terms of Actual Infinity) there is a Gödel number of an axiom (some x) that proves G (some xU{x}) in A, which is a contradiction in A. Therefore, A is inconsistent.


Model 2:

Let any x or any xU{x} be axioms (wffs that are not proven) in A.

G axiom states: "There is no number m such that m is the Gödel number of a proof in A, of G"

Since G is already an axiom in A (where A is an infinite set of axioms, such that Infinity is taken in terms of Actual Infinity) it is actually a wff that is true in A, which does not have any Gödel number that is used in order to encode G's proof (since axioms are true wffs that do not need any proof in A).

But then no proof is needed and mathematicians are out of job (therefore it is an unwanted solution).

---------------------------

So, in both models infinitely in terms of Actual Infinity (an infinite set that is taken as a complete whole) does not establish an interesting formal system.

---------------------------

Please pay attention to the following remarks, before you reply:

Since A is a set of infinitely many wffs that are taken as a complete whole (this is exactly what Actual Infinity is about) there cannot be a Gödel number that is not already in A, whether whether some wff is an axiom or a theorem in A (see Model 1). So, one can't use G as a wff that is unproven in A, as done in case of GIT, since if one does this, one deduces in terms of Potential Infinity, which is not a part of my argumnt.

 

[jsfisher]

You just need a function that determines whether x is a member of A. You don't have one.

x → xU{x} is the bijective membership function of A.

It is neither bijective nor a membership function. A membership functions are true/false-valued.

 

[doronshadmi]

It is neither bijective nor a membership function. A membership functions are true/false-valued.

x → xU{x} is bijective, for example:

{} → {{}}
{{}} → {{},{{}}}
{{},{{}}} → {{},{{}},{{},{{}}}}
... etc.

and also determine the members of A (exactly as the members of an inductive set are determined by The Axiom Of Infinity).

 

[jsfisher]

x → xU{x} is bijective

Repeating your false claim doesn't make it true. Nothing maps to the empty set.

Be that as it may, it isn't a membership function, either, since it doesn't answer the question, "Is m a member of set A?" F(x) = x U {x} doesn't.

and also determine the members of A (exactly as the members of an inductive set are determined by The Axiom Of Infinity).

The Axiom of Infinity doesn't identify any particular set; it doesn't provide a membership function. It merely states two properties the set has (i.e. that it contains the empty set and that every member of the set also has its successor as a member).

The Axiom is silent on whether, for example, {{{{ }}}} is in the set.

The Axiom must to be coupled with other axioms to conclude von Neumann's ordinal is a set in ZF.

 

[doronshadmi]

Repeating your false claim doesn't make it true. Nothing maps to the empty set.

The Axiom of Infinity doesn't identify any particular set; it doesn't provide a membership function.

Repeating your false claim doesn't make it true. As for the empty set, it is a member of A (as given below) and nothing maps to the empty set since it is a domain object (as seen in http://www.internationalskeptics.com/forums/showpost.php?p=12806505&postcount=3418).

There is set A (the set which is postulated to be infinite), such that the empty set is a member of A, and such that whenever any x is a member of A, the set formed by taking the union of x with its singleton {x}, is also a member of A.

So, both x and xU{x} are members of A.

So, Syntactically x → xU{x} is the bijective membership function of A (no x, xU{x} or any gödel number m are missing from A).

Be that as it may, it isn't a membership function, either, since it doesn't answer the question, "Is m a member of set A?" F(x) = x U {x} doesn't.

m is a Gödel number of a proof of some xU{x} in A (according to model 1 (seen in http://www.internationalskeptics.com/forums/showpost.php?p=12804530&postcount=3416)) exactly because A is determined in terms of Actual Infinity as a complete whole (no m, x or xU{x} are missing from A).

The Axiom of Infinity doesn't identify any particular set; it doesn't provide a membership function. It merely states two properties the set has (i.e. that it contains the empty set and that every member of the set also has its successor as a member).

The Axiom is silent on whether, for example, {{{{ }}}} is in the set.

I am not talking about ZF, but about the construction of A such that Infinity is taken as a complete whole (no x, xU{x} or any gödel number m are missing from A).

If you don't like xU{x}, it can easily be replaced by {x} as follows:

{} → {{}}
{{}} → {{{}}}
{{{}}} → {{{{}}}}}
... etc.

For example, syntactically (by formalism without semantics) there is set A (the set which is postulated to be infinite), such that the empty set is a member of A, and such that whenever any x is a member of A, the set {x}, is also a member of A.

So, both x and {x} are members of A.

So, Syntactically x → {x} is the bijective membership function of A (no x, {x} or any gödel number m are missing from A).