Cont: Deeper than primes - Continuation 2

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I don't get the concept of "missing set members". A set member is always there. It can't call in sick or miss the bus or anything. At least that's what I used to believe until Doron set things "straight".
 
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doronshadmi said:
In case that you are still missing it, all it matters is that Infinitely many things are taken in terms of Platonic Infinity, which means that such collection is complete and therefore inconsistent by GIT.
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You continue to conflate Cantor's meaning of "complete" with Godel's. Please stop that. They are completely different concepts.
 
Minor nitpick:

doronshadmi said:
No one of A members, which are encoded by Godel numbers (which are actually the set of all natural numbers) is missing.
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To my understanding, this quote is wrong.

Gödel numberingWP "... is a function that assigns to each symbol and well-formed formula of some formal language a unique natural number, called its Gödel number."

I can use only odd numbers, numbers divisible by four, numbers with even number of syllables when speaking them in Klingon, or any other arbitrary rule for Godel numbers. That is *not* the set of all natural numbers.
 
Little 10 Toes said:
Minor nitpick:



To my understanding, this quote is wrong.

Gödel numberingWP "... is a function that assigns to each symbol and well-formed formula of some formal language a unique natural number, called its Gödel number."

I can use only odd numbers, numbers divisible by four, numbers with even number of syllables when speaking them in Klingon, or any other arbitrary rule for Godel numbers. That is *not* the set of all natural numbers.
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You indeed can use a set of different rules, but this set is complete exactly as the set of all natural numbers is complete if Infinity is taken in terms of Platonic Infinity (please carefully observe http://www.internationalskeptics.com/forums/showpost.php?p=12781487&postcount=3340 including its link).
 
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jsfisher said:
They are completely different concepts.
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Please support your argument.

jsfisher said:
The Axiom of Infinity at no point says that "no member is missing"
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Also please be aware of the following:

1) If ZF(C) Axiom Of Infinity is not necessarily taken in terms of Platonic Infinity, then ZF(C) Axiom Of Infinity is taken in terms of Platonic Infinity OR Not (useless tautology).

2) If ZF(C) Axiom Of Infinity is not necessarily taken in terms of Platonic Infinity, then it can't be used in order to establish even the set of all natural numbers (which means that N (and |N|) is not necessarily established ZF(C)).
 
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Some correction of my previous post, the end of it has to be:

(which means that N (and |N|) is not necessarily established by ZF(C)).
 
doronshadmi said:
Please support your argument.
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It is a key to your argument, not mine. You need to show that Cantor completeness and Godel completeness refer to the same thing.

They don't, but you will need to figure that out on your own.
 
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doronshadmi said:
(Also this time please do not ignore all of what is written in http://www.internationalskeptics.com/forums/showpost.php?p=12782645&postcount=3346 including your quotes, and my correction in http://www.internationalskeptics.com/forums/showpost.php?p=12782699&postcount=3347).
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Most of what you post should be ignored. Little of what you write follows any sort of logical argument, and you don't even bother to understand what most of the terms you use actually mean. "Completeness" is just the latest example.

At least take the time to find out what completeness of a formal system means.
 
jsfisher said:
Most of what you post should be ignored. Little of what you write follows any sort of logical argument, and you don't even bother to understand what most of the terms you use actually mean. "Completeness" is just the latest example.

At least take the time to find out what completeness of a formal system means.
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Happy birthday jsfisher.

You did not support your argument, please this time support it.

Thank you.
 
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jsfisher said:
At least take the time to find out what completeness of a formal system means.
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Here is the relevant parts taken from Wikipedia:
Syntactical completeness

Since A formal system S is syntactically complete or deductively complete or maximally complete if for each sentence (closed formula) φ of the language of the system either φ or ¬φ is a theorem of S. This is also called negation completeness, and is stronger than semantic completeness.

In another sense, a formal system is syntactically complete if and only if no unprovable sentence can be added to it without introducing an inconsistency.

...

Gödel's incompleteness theorem shows that any recursive system that is sufficiently powerful, such as Peano arithmetic, cannot be both consistent and syntactically complete.
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Since A is syntactically complete in terms of Platonic Infinity (where no process is involved), it is inconsistent by GIT, as follows:

G (which is wff in A and therefore has a Godel number) states: "There is no number m such that m is the Godel number of a proof in A, of G" (in short: G, which is wff in A, is not provable in A).

Since A is syntactically complete in terms of Platonic Infinity (A is an extended ZF(C) by using ZF(C) Axion Of Infinity on ZF(C) itself, where Infinity is taken in terms of Platonic Infinity) there is no Godel number that is not already used in A in order to encode every possible wff in A (where wff G in A is not exceptional).

So there is G' (which is wff in A and therefore has a Godel number) that proves the non-provability of G (which is wff in A and therefore has a Godel number) in A, which is a contradiction.

In other words, A is syntactically complete in terms of Platonic Infinity and therefore inconsistent by GIT.
 
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jsfisher said:
In what way? Your set, A, is the set of natural numbers. It is not a formal system.
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No, it is equivalent to the set of all natural numbers.

A is a complete formal system such that no wff is missing, exactly as no natural number is missing from the set of all natural numbers.
 
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doronshadmi said:
No, it is equivalent to the set of all natural numbers.
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That would make it equivalent to the set of natural numbers, not any formal system.

A is a complete formal system such that any wff is not missing
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...along with a whole bunch of things that aren't well-formed statements (however they might be encoded). Moreover, you have included statements along with their negation, so whatever formal system you imagine yourself to have constructed, it is inconsistent by its very construction.

And it is not related to ZF nor ZFC, so any link from your bogus conclusion on completeness doesn't connect to ZF or ZFC Set Theory (which are formal systems).
 
jsfisher said:
...along with a whole bunch of things that aren't well-formed statements (however they might be encoded).
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They are encoded as non-wff in A and therefore are ignored.

Yet all wff are in A exactly because there is no Godel number that is not in A if Infinity is taken in terms of Platonic Infinity.

jsfisher said:
Moreover, you have included statements along with their negation, so whatever formal system you imagine yourself to have constructed, it is inconsistent by its very construction.
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This is exactly the result if all Godel numbers are already used in A (to encode non-wff (which are ignored) or wff (which are not ignored)).

Let's use only G as a wff.

G (which is wff in A and therefore has a Godel number in A that is not ignored) states: "There is no number m such that m is the Godel number of a proof in A, of G" (in short: G, which is wff in A, is not provable in A).

Since Godel number of G in A is not ignored in A, it is actually used in order to prove G statement, which is a contradiction in A.

"In another sense, a formal system is syntactically complete if and only if no unprovable sentence can be added to it without introducing an inconsistency."

Since A is defined in terms of Platonic Infinity (everything is already included in A) unprovable sentence is already provable in A.

jsfisher said:
And it is not related to ZF nor ZFC, so any link from your bogus conclusion on completeness doesn't connect to ZF or ZFC Set Theory (which are formal systems).
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A is an extension of ZF(C) in terms of Platonic Infinity, so it is complete and therefore inconstant by GIT.

Any attempt to establish ZF(C) by using ZF(C) Axiom Of Infinity on ZF(C) not in terms Platonic Infinity, actually rejects the existence of infinite and complete collection (for example, one actually rejects the existence of the set all natural numbers).
 
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jsfisher said:
Most of what you post should be ignored. Little of what you write follows any sort of logical argument, and you don't even bother to understand what most of the terms you use actually mean. "Completeness" is just the latest example.

At least take the time to find out what completeness of a formal system means.
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Wow, this thread is still going - after 10 years - and Doron isn't making any more sense than he was then.

I admire your patience and tenacity - although I have to wonder why you still do it ;)
 
dlorde said:
Wow, this thread is still going - after 10 years - and Doron isn't making any more sense than he was then.

I admire your patience and tenacity - although I have to wonder why you still do it ;)
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It was a weak moment. It is once again proven pointless, though. Doronshadmi is digging in to defend his ignorance. He will not be shaken, stirred, bent, folded, spindled, or mutilated.
 
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