Deeper than primes - Continuation

[jsfisher]

No. Look up the word enables. Oh, and previews.

The fact you are so careless with language says a lot.

On the one hand, since I do not think English is Doron's first language, I cut him some slack; on the other, though, since he often stubbornly adheres to his linguistic nonsense despite correction, the slack is extremely limited.

In any language, his expressions are muddled.

In other words, I love apples.

 

[laca]

Published anything yet, doron? You could spare us until there's been some actual peer review of your "work", don't you think? You fed us enough gibberish and I personally am fully convinced you're able to do so endlessly. No point in doing it over and over.

 

[doronshadmi]

If I recall correctly, both IZF and CZF include intuitionist versions of the Axiom of Infinity, suggesting broad acceptance, not rejection, of actual infinity.

I agree with you jsfisher, we should not talk about acceptance or rejection of Actual Infinity, but about the understanding of the notion of Actual Infinity.

The first step in order to understand this notion is to ask: "What is Actual Infinity?"

My answer is:

Actual Infinity is the existence that is logically always true (tautological existence).

IZF, CZF or ZF are axiomatic frameworks about sets, so in order to understand Actual Infinity in therms of sets, we first have to understand what is a set in terms of existence, where tautological existence and non-tautological existence are (a) and (b), as follows:

a) The discovered platonic (and therefore objective) level of existence, where this level of existence is logically a tautology (existence is always true) (this notion is symbolized by the outer "{" and "}" of any given set, whether it is empty or non-empty).

b) The invented non-platonic (and therefore subjective) level of existence, where this level of existence is logically not a tautology (existence is not always true) (this notion is defined between the outer "{" and "}" of any given set, and its non-tautological existence is symbolized (in the case of non-empty set) or not symbolized (in the case of the empty set)).

In other words, as long as IZF, CZF or ZF axioms are used in order to understand Actual Infinity in terms of (b), no understanding of Actual Infinity is established.

Here is, for example, ZF Axiom Of Infinity:

"There exists a set X (this is (a) part of the axiom) such that (this is (b) part of the axiom) {} is a member of X and, whenever a set y is a member of X, then S is also a member of X."

By this axiom, X has two levels of infinity, where (a) is the actual level and (b) is the non-actual level.

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

A request like "Please show me a member of set X that is missing from X"
(where the aim of this request is to support the notion of completeness (and therefore Actual Infinity) at
the non-platonic level of existence of members) is irrelevant if one
understands the different levels of existence of set X, by using Philosophy
(as meta-view of Mathematics) and Mathematics (the non-platonic level of
existence of the members of set X is inaccessible to the platonic level of
existence of set X).

 

[jsfisher]

I agree with you jsfisher, we should not talk about acceptance or rejection of Actual Infinity, but about the understanding of the notion of Actual Infinity.

Great! except that isn't what I said.

The first step in order to understand this notion is to ask: "What is Actual Infinity?"

My answer is:

Actual Infinity is the existence that is logically always true (tautological existence).

My, isn't that curious. You take a perfectly fine, well-understood concept of completeness and infinitely many as in, for example, the set of integers, and you decide you can redefine it to mean something entirely different.

No. Your answer is horribly wrong.

IZF, CZF or ZF are axiomatic frameworks about sets...

...each built on a philosophic framework that isn't Doron's to redefine post hoc.

 

[doronshadmi]

No. Look up the word enables. Oh, and previews.

The fact you are so careless with language says a lot.

It is of course previous. English is not my mom's language so I use a speller.

From time to time I write a word in a wrong way and the speller automatically changes it to some valid word, even if it does not fit to what I actually wish to express. If the result of the automatic correction looks like the word that I actually wish to express, there is a chance that I'll miss the mistake.

 

[doronshadmi]

You take a perfectly fine, well-understood concept of completeness and infinitely many as in, for example, the set of integers, and you decide you can redefine it to mean something entirely different.[/I].

It is not entirely different it is simply refined, and you simply do not accept this refining, so?

 

[jsfisher]

It is not entirely different it is simply refined, and you simply do not accept this refining, so?

No, and no.

It is entirely different, and it is your leave to redefine I reject.

 

[doronshadmi]

It is entirely different,

Unlike you, jsfisher, I explicitly define Actual Infinity and Non-actual Infinity in terms of existence, by using Logic:

Actual Infinity is the existence that is logically always true (tautological existence).

From this definition it is easily deduced that Non-actual Infinity is the existence that is logically not always true (it is not tautological existence).

By using this definitions I explicitly demonstrate how they are used by ZF Axiom Of Infinity.

In other words, jsfisher, you have no case (all you have is an orthodox standpoint about my definitions and their use).

 

[realpaladin]

Unlike you, jsfisher, I explicitly define Actual Infinity and Non-actual Infinity in terms of existence, by using Logic:

Actual Infinity is the existence that is logically always true (tautological existence).

A singleton is infinite????

 

[jsfisher]

Unlike you, jsfisher, I explicitly define Actual Infinity and Non-actual Infinity in terms of existence, by using Logic

Redefinition of established terminology is not within you purview.

...
In other words....

Yes, yes, your love for apples is well-established.

 

[doronshadmi]

A singleton has cardinality 1, for example {|{}|} or {|{x}|} or {|x|} (where by ZF x is a set).

Again, given any set, the outer "{" and "}" are used to notate the notion of the level of existence that is logically always true.

This is not the case with the level of members (they may exist (for example {{}}) or not (for example {}).

 

[doronshadmi]

Redefinition of established terminology is not within you purview.

Your used reasoning is not sufficient in order to distinguish between tautological existence and non-tautological existence, so?

 

[jsfisher]

A singleton has cardinality 1, for example {|{}|} or {|{x}|} or {|x|} (where by ZF x is a set).

You don't get to redefine established notation, either.

 

[doronshadmi]

You don't get to redefine established notation, either.

Notations (or their absence) are used in order to express notions.

This is exactly what is done in order to distinguish between tautological and non-tautological levels of existence, which explicitly used by ZF Axiom Of Infinity (more details are given in http://www.internationalskeptics.com/forums/showpost.php?p=10039964&postcount=3903).

jsfisher, your orthodox reasoning is not useful in order to address the difference between tautological and non-tautological levels of existence, where these levels are explicitly used by ZF Axiom Of Infinity.

In other words, you have no case (except some fetish attitude about notations).

 

[doronshadmi]

Redefinition of established terminology is not within you purview.

Redefinition of orthodox terminology is not within the scope of orthodox terminology, so?

 

[doronshadmi]

You take a perfectly fine, well-understood concept of completeness and infinitely many as in, for example, the set of integers, and you decide you can redefine it to mean something entirely different.[/I].

Any set, including the set of integers, is the result of the linkage between two level of existence, where one level is logically tautological existence, and the other level is logically non-tautological existence.

The use of these two levels of existence is clearly shown in ZF axiomatic framework (http://www.internationalskeptics.com/forums/showpost.php?p=10039964&postcount=3903), where your orthodox reasoning is insufficient in order to state any useful thing about it.

 

[jsfisher]

Any set, including the set of integers, is the result of the linkage between two level of existence, where one level is logically tautological existence, and the other level is logically non-tautological existence.

Or so you claim. Corrupting language is not an acceptable method for establishing your claim.

The use of these two levels of existence is clearly shown in ZF axiomatic framework

So clearly, in fact, that you are completely unable to show where these levels exist in ZF. Instead, you simply reiterate your claim, or re-post links to previous claim reiterations, or both. Never do you actually show anything other than a baseless claim infused with gibberish.

 

[doronshadmi]

Or so you claim. Corrupting language is not an acceptable method for establishing your claim.



So clearly, in fact, that you are completely unable to show where these levels exist in ZF. Instead, you simply reiterate your claim, or re-post links to previous claim reiterations, or both. Never do you actually show anything other than a baseless claim infused with gibberish.

jsfisher, from your orthodox standpoint the best you can get about http://www.internationalskeptics.com/forums/showpost.php?p=10039964&postcount=3903 is indeed gibberish.

I wish you the best.

 

[Little 10 Toes]

"There exists a set X (this is the part that uses set's platonic level of existence) such that (this is the part that uses set's non-platonic level of existence (the level of members, which defines set's identity, but not set's platonic level of existence)) {} is a member of X and, whenever a set y is a member of X, then S is also a member of X."

"There exists a set X (this is the part that uses set's platonic level of existence) such that (this is the part that uses set's non-platonic level of existence (the level of members, which defines set's identity, but not set's platonic level of existence)) {} is a member of X and, whenever a set y is a member of X, then S is also a member of X."

Still waiting for the citation that shows exactly your quote.

 

[doronshadmi]

Still waiting for the citation that shows exactly your quote.

This is my last post to you on this subject.

Please look at http://www.internationalskeptics.com/forums/showpost.php?p=10039964&postcount=3903.

Now take

"There exists a set X (this is (a) part of the axiom) such that (this is (b) part of the axiom) {} is a member of X and, whenever a set y is a member of X, then S is also a member of X."

and omit what is written in italic letters within the brackets

( http://en.wikipedia.org/wiki/Zermelo–Fraenkel_set_theory#7._Axiom_of_infinity )
... there exists a set X such that the empty set Ø is a member of X and, whenever a set y is a member of X, then S is also a member of X.

and here it is.