Deeper than primes - Continuation

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doronshadmi said:
At the level that I am talking about, Traditional mathematics does not use a function without an inverse (which means, a function that has an input, but it does not have any output).
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Good. Now we are getting somewhere. You are stating here that your inverse has nothing to do with the traditional mathematical definition of inverse.

So, as an analogy: "At the level I am playing chess, a cow does not have a clue about mate."

EDIT: Deleted the explanation for Doron. Too much anticipatory fun watching him butcher it...

doronshadmi said:
You simply have problems to grasp http://www.internationalskeptics.com/forums/showpost.php?p=9833876&postcount=3640 or http://www.internationalskeptics.com/forums/showpost.php?p=9837661&postcount=3659
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You simply can't explain it.


doronshadmi said:
Your are a liar (you are actually lying to yourself), because this is exactly what I have said in http://www.internationalskeptics.com/forums/showpost.php?p=9837661&postcount=3659 or http://www.internationalskeptics.com/forums/showpost.php?p=9833876&postcount=3640.
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No Doron, don't try your usual sneaky tactic of posting zillions of posts in between so it looks like 'you have always said that'.

You did not, and you make your sneakiness worse by acting as if you did.

Now stop acting like a jester and get on with the show.
 
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doronshadmi said:
I are wrong.

The level of function that I am talking about is more straightforward than the level of Traditional mathematics.
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Well, <SNIP> your function is not straight at all, since you need two points and you only have 1.


Edited by LashL: 
Edited to remove breach. Remain civil, please.
 
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Edited for breach of Rule 0 and Rule 12. Remain civil, please, and remember to address the argument rather than attacking the arguer.
Replying to this modbox in thread will be off topic  Posted By: LashL
 
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sympathic said:
Stop inventing your stuff and calling it with the same name as something that is already well defined in Math.
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Math is not a privet property of any particular school of thought, so stop trying to force what the community of traditional mathematicians calls well-defined.
 
realpaladin said:
Well, <SNIP>, your function is not straight at all, since you need two points and you only have 1.
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<SNIP>, a function starts also from 1 point, but in this case it simply does not have an inverse function.

Edited by LashL: 
Edited to remove breach. Remain civil, please.
 
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doronshadmi said:
Math is not a privet property of any particular school of thought, so stop trying to force what the community of traditional mathematicians calls well-defined.
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This has nothing to do with Math being private or any school of thought. It is the same as referring to the colour of the sky as "green". You can go ahead and use the same name for well established notions when you actually mean something else. In addition to it being plain wrong, it makes communication with you even more difficult than it already is.
 
sympathic said:
This has nothing to do with Math being private or any school of thought. It is the same as referring to the colour of the sky as "green".
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EDIT:

It is the same as restricting the colour of the sky as only-blue (= function must be based on pairs, in order to be defined) but the colour of the sky is not only-blue (= a function is defined also if it is based on a singleton).

By using such a function in a mathematical mutiverse (which is a generalization the traditional mathematical universe (exactly as multiset is a generalization of set), one easily deduces that the notion of transfinite cardinality does not hold water (http://www.internationalskeptics.com/forums/showpost.php?p=9823863&postcount=3563 , http://www.internationalskeptics.com/forums/showpost.php?p=9830197&postcount=3609).

You simply ignore http://www.internationalskeptics.com/forums/showpost.php?p=9828130&postcount=3602 like any good traditional mathematician.
 
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doronshadmi said:
I are wrong.
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You will get no argument on that from anyone here.

The level of function that I am talking about is more straightforward than the level of Traditional mathematics.
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No, you were not talking about functions at all. You were just making stuff up as you frequently do to cover your ignorance and your blunders.

You don't get to declare Mathematics wrong by redefining things. That's not how it works. Feel free to invent all the nonsense you like for doronetics -- and it all has been nonsense -- but you get no sway over Mathematics.

Your claim about functions without inverses was fantastically wrong. You know it was wrong.

<Edited to remove a least civil part>
 
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doronshadmi said:
EDIT:

Because it is the signature the non-entropic conditions of the mathematical multiverse (http://www.internationalskeptics.com/forums/showpost.php?p=9830197&postcount=3609, http://www.internationalskeptics.com/forums/showpost.php?p=9824985&postcount=3569).
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And this is of what significance to whom and why?

I have not seen a single thing produced by OM except for irritation (proof is in the yellow cards) and strife.

So, please explain what makes this better than traditional science which gives us beauty, harmony and understanding.

After all, you claim in your PDF's that OM will bring unity; you made that claim in the two islands thought experiment more expressly and visually.

Now then, care to show how exactly that is of any importance at all?
 
In a few posts it is time again for a recap of all the errors, including the non-rational tactics employed.

If there is anyone, besides Doron, that has seen anything that has a relation to mathematics, please mention so, otherwise the recap will need to point out that we are working here with a personal hobby construct and not mathematics.

This seems important to me as traditional mathematics is referenced a lot, and most people, including me at some point, make the mistake of thinking that the discussion is about mathematics.
 
sympathic said:
This has nothing to do with Math being private or any school of thought. It is the same as referring to the colour of the sky as "green". You can go ahead and use the same name for well established notions when you actually mean something else. In addition to it being plain wrong, it makes communication with you even more difficult than it already is.
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Actually, the sky is green (in Japanese).
 
doronshadmi said:
You simply ignore all of what is written in http://www.internationalskeptics.com/forums/showpost.php?p=9839144&postcount=3691.
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Yes we do. We know it.

The thing is, what is written there has neither bearing on philosophy nor on real mathematics.

All of us are waiting for some real insights, backed by a rigorously proven framework.

As long as claims are staying in the realm of "You don't get it." or boxed-in smileys we feel treated in an uncivilized way and we feel that you attack the arguer not the argument.

What all of us are really waiting for is:

- Either something that is truly insightful (and not because just one person says it is; a lot of stuff on the homeshopping channel would qualify).

- Or a chain of reasoning (details need not be filled in, just a roadmap) that leads us from the proposed unity to peace, as you described in your answer to the two islands thought experiment.

- Or a correct proof (not a claim, please do not confuse them) that you can do something with OM that can not be done with traditional mathematics, but in the realm of practical application.

So while we wait, we amuse ourselves a bit with social interaction.
 
In his very interesting article http://www.austms.org.au/Gazette/2006/Jul06/jcrossley.pdf Prof. John Newsome Crossley writes: "Mathematics has no end, and many ends".

I wish to share my view of the, so called, paradox.

Cantor defined the notion of actual infinity by providing the transfinite mathematical universe, which uses accurate measurement units (transfinite cardinality like aleph-0 etc.) of the number of members of sets.

Godel was a Platonist, which gave him the motivation to prove that no axiomatic mathematical system that is strong enough to deal with Arithmetic, can be both complete (can prove its all true statements within its own framework) and consistent (avoiding contradictions). In other words, he wished to prove that the symbolic formal approach is insufficient in order to fully express (fully capture) the Platonic realm (according to Godel's view, the symbolic formal approach is subjective with respects to the objectivity of the Platonic realm).

By following Godel approach, I wish to extend the notion of symbolic formal approach, such that it will be abstractly understood both by verbal_symbolic AND visual_spatial mathematician's skills.

For example, {} is a mathematical expression of the, so called, empty set.

The cardinality of a given set is defined by the number of its members, so in the case of {} the cardinality is |{}| = 0.

0, in this case, means that the outer "{" and "}" or the void between {} expression, are not in the scope of (abstract or non-abstract) existing members (more precisely, "{" and "}" is "above" existing members, and the void between {} expression is "below" existing members).

In other words, we have used here an abstraction that combines verbal_symbolic AND visual_spatial mathematician's skills in order to understand {}.

By following this abstract approach one immediately understands that the inner {} of {{}} expression, is not extensible into the outer "{" and "}" AND not reducible into the void between the inner {} expression, and so is the case about {…{…}…} expression (the inaccessibility of the number of levels of (abstract or non-abstract) existing members, to the Platonic realm).

{{}} or {…{…}…} demonstrate that given mathematical universe, the actually infinite (expressed by the outer "{" and "}") or the actually finite ("expressed" by the void between {} expression) are inaccessible to any number of (abstract or non-abstract) existing members.

By following this notion one immediately understands that the Cantorean notion of actual infinity, which is based on the notions of accurate values of the number of members (aleph-0, etc.), is not satisfied.

In other words, the Platonic realm in itself is "above" and "blow" members, where the level of members is inaccessible to it.

So, at the level of members (or more generally, the level of collections) "Mathematics has no end, and many ends", but it is not a paradox if one uses his\her verbal_symbolic AND visual_spatial skills, in order to be aware of the Platonic realm w.r.t the level of collections (which means that the mathematician's awareness is a significant factor of the mathematical research and its (abstract or non-abstract) results, unlike the "hard" Platonic view).
 
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By ZFC axiom of infinity "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" ( http://en.wikipedia.org/wiki/Axiom_of_infinity ).

According to this axiom we get many finite sets, where there is no the greatest set that is the union of x with its singleton {x}, as a member of I.

Transfinite cardinality is derived from the notion that there is an accurate size to the number of the members of I, such that this accurate size is greater than the cardinality of any given set that is a member of I.

Since there is no the greatest set that is the union of x with its singleton {x} as a member of I, there are always more members of I that are inaccessible to any accurate size, or in other words, the notion of transfinite cardinality is not satisfied.
 
doronshadmi said:
By ZFC axiom of infinity "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" ( http://en.wikipedia.org/wiki/Axiom_of_infinity ).

According to this axiom we get many finite sets, where there is no the greatest set that is the union of x with its singleton {x}, as a member of I.
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No, all the other axioms of ZFC Set Theory guarantee us many finite sets. The Axiom of Infinity stipulates the existence of an infinite set.

Transfinite cardinality is derived from the notion that there is an accurate size to the number of the members of I, such that this accurate size is greater than the cardinality of any given set that is a member of I.
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No, it isn't. In fact "size" in a numerical sense as you mean it is completely absent from base ZFC.

Since there is no the greatest set that is the union of x with its singleton {x} as a member of I, there are always more members of I that are inaccessible to any accurate size, or in other words, the notion of transfinite cardinality is not satisfied.
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The problem with founding an argument on false assumptions, is that you usually end up with a false conclusion. Case in point.

Doron, all you'd need to do to make your point is prove within the confines of ZFC that |P(I)| = |I|. These bogus arguments of yours just aren't cutting it.
 
Little 10 Toes said:
No
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No to your no.

You are invited to abstractly show exactly how the outer "{" and "}" or the void between {} expression, are in the scope of (abstract or non-abstract) existing members, if you really wish to support your "no".
 
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jsfisher said:
No, all the other axioms of ZFC Set Theory guarantee us many finite sets. The Axiom of Infinity stipulates the existence of an infinite set.
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No jsfisher, it does not stipulate the existence of an infinite set in the actual sense, exactly because (by using this axiom) the construction of I members in terms of the union of x with its singleton {x}, guarantees the absence of the greatest member of I, and as a result |I| accuracy is not satisfied within ZFC.

This is exactly what is needed in order to conclude that actual infinity is not satisfied in terms of the number of members of a given non-empty set, and it is equivalent to the abstract notion that the outer "{" and "}" are inaccessible to any number of members within them.

Moreover, the abstract notion that aleph0 = aleph0+1 etc., is equivalent to the abstract notion that are no outer "{" and "}" to the outer "{" and "}" (otherwise they can't be considered as outer "{" and "}", in the first place).

In other words, an accurate value like aleph0 is equivalent to the abstract notion of the accuracy of "{" and "}" as being absolutely outer w.r.t and number of number of members within them.

Cantor's mistake and his followers, it the attempt to formalize this accuracy at the level of the abstract notion of members.

More details of this approach are given in http://www.internationalskeptics.com/forums/showpost.php?p=9980885&postcount=3702.
 
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Let's correct some typo mistakes of my previous post.

Instead of

doronshadmi said:
Moreover, the abstract notion that aleph0 = aleph0+1 etc., is equivalent to the abstract notion that are no outer "{" and "}" to the outer "{" and "}" (otherwise they can't be considered as outer "{" and "}", in the first place).

In other words, an accurate value like aleph0 is equivalent to the abstract notion of the accuracy of "{" and "}" as being absolutely outer w.r.t and number of number of members within them.
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It has to be:

Moreover, the abstract notion that aleph0 = aleph0+1 etc., is equivalent to the abstract notion that there are no outer "{" and "}" to the outer "{" and "}" (otherwise they can't be considered as outer "{" and "}", in the first place).

In other words, an accurate value like aleph0 is equivalent to the abstract notion of the accuracy of "{" and "}" as being absolutely outer w.r.t any number of members within them.

By understanding the inaccessibility of any number of members to aleph0 or to the outer "{" and "}", we also understand that an argument like "no I member is missing", does not change this inaccessibility.

So, a phrase like "show me a member of set I that is missing from I" has no impact on the inaccessibility of |I| to the accuracy of aleph0, or in other words, |I| < aleph0.
 
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doronshadmi said:
No jsfisher, it does not stipulate the existence of an infinite set in the actual sense, exactly because (by using this axiom) the construction of I members in terms of the union of x with its singleton {x}, guarantees the absence of the greatest member of I, and as a result |I| accuracy is not satisfied within ZFC.
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Reading comprehension issues again? The Axiom of Infinity does not provide a construction; it stipulates an existence. Look carefully at the wording of the Axiom, "there exists a set X such that....":

04a649153aa8f4e3aa90ea6ac91d13ff.png


Imagining the Axiom to be anything else is just your, well, imagination.
 
jsfisher said:
Reading comprehension issues again? The Axiom of Infinity does not provide a construction; it stipulates an existence. Look carefully at the wording of the Axiom, "there exists a set X such that....":

[qimg]http://upload.wikimedia.org/math/0/4/a/04a649153aa8f4e3aa90ea6ac91d13ff.png[/qimg]

Imagining the Axiom to be anything else is just your, well, imagination.
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"there exists a set X such that....", where what comes after such that is a construction method of X members, which guarantees the absence of the greatest member of X, and as a result |X| accuracy is not satisfied within ZFC, as shown in http://www.internationalskeptics.com/forums/showpost.php?p=10011736&postcount=3708.

Imagining this axiom as a guarantee that |X| = aleph0 is just your, well, imagination (where "your" is Cantor's followers).
 
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doronshadmi said:
"there exists a set X such that....", where what comes after such that is a construction method of X members...
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No. That is not how predicate logic works. We have a stipulation of existence (that being the "there exists" part), and we have conditions met by that thing that exists (that being the "such that" part).

Doron, if you cannot understand Mathematics and mathematical expression, then you are in a very poor position to comment on its correctness.
 
jsfisher said:
... we have conditions met by that thing that exists (that being the "such that" part).
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EDIT:

jsfisher, by these conditions (as found by this axiom) |X| is inaccurate such that it is < aleph0, which is accurate.

Cantor's reasoning (and yours, as one of his followers) fails to understand the meaning of aleph0 = aleph0+1 and its equivalence to the outer "{" and "}" , as shown in http://www.internationalskeptics.com/forums/showpost.php?p=10011736&postcount=3708.

In other words:

[qimg]http://upload.wikimedia.org/math/0/4/a/04a649153aa8f4e3aa90ea6ac91d13ff.png[/qimg] (also see http://en.wikipedia.org/wiki/Axiom_of_infinity#Formal_statement)

(there exists a set X such that the empty set {} is a member of X and, whenever a set y is a member of X, then S(y) is also a member of X)

(see http://en.wikipedia.org/wiki/Zermelo–Fraenkel_set_theory#7._Axiom_of_infinity)

guarantees that the number of X members is inherently under construction ("and, whenever a set y is a member of X, then S(y) is also a member of X"), and as a result it does not have the property of aleph0 = aleph0+1, which is a property of that is not under construction exactly as the outer "{" and "}" are not under construction (otherwise they can't be considered as the outer "{" and "}", in the first place).
 
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doronshadmi said:
EDIT:

jsfisher, by these conditions (as found by this axiom) |X| is inaccurate such that it is < aleph0, which is accurate.
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No. The Axiom of Infinity makes no mention of cardinality nor of any of the transfinite numbers.

Cantor's reasoning (and yours, as one of his followers) fails to understand the meaning of aleph0 = aleph0+1 and its equivalence to the outer "{" and "}"
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Gibberish.

However, if there is a defect in Cantor's Theorem, then you should be able to clearly demonstrate it within the bounds of ZFC Set Theory. Please, have at it.

As is your style, you assert but never prove, so I will not hold my breath waiting for this demonstration.
 
jsfisher said:
No. The Axiom of Infinity makes no mention of cardinality nor of any of the transfinite numbers.
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In that case, jsfisher, what is the number of members of set X, according to The Axiom of Infinity?

jsfisher said:
However, if there is a defect in Cantor's Theorem, then you should be able to clearly demonstrate it within the bounds of ZFC Set Theory. Please, have at it.
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Already done in http://www.internationalskeptics.com/forums/showpost.php?p=10013503&postcount=3712.

In other words, the number of members of set X (according to The Axiom of Infinity) < aleph0, exactly because (according to The Axiom of Infinity) the number of members of set X is inherently under construction ("and, whenever a set y is a member of X, then S(y) is also a member of X").

Your reasoning is not capable to understand the inaccessibility of the number of members of set X to aleph0 (which is at the level of the outer "{" and "}") exactly as it is not capable to understand the inaccessibility of the number of members of set X to the outer "{" and "}", which is the level above being a member, of any given set.

This inaccessible level is a property of any given set, whether it is empty, or not, for example:

|{}|=0 , |{{}}|=1 |{{},{{}}}|=2 etc. , or |{{}, {{}}, {{},{{}}}, ... }| < aleph0 exactly because no number of members of any given set is accessible to the level above being a member (where this abstract notion (that is not understood by your naive reasoning about sets, and therefore called by you "Gibberish") is notated by the outer "{" and "}").

As about infinity, your naive reasoning about sets unable to distinguish between potential infinity (which is at the level of members) and actual infinity (which is above the level of members (and notated here by the outer "{" and "}")).
 
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doronshadmi said:
In that case, jsfisher, what is the number of members of set X, according to The Axiom of Infinity?
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Number? I see you are still confused about cardinality and all that. Set Theory does not deal with numbers. That would be Arithmetic. Don't you remember from the last time you demonstrated your confusion? At the set-theoretic level, cardinality is a relative measure.

If you want to displace Cantor and ravage ZFC Set Theory, you'll need to expose the weaknesses within ZFC. Cantor's Theorem relies on the Axiom of Infinity (an axiom), the Axiom of the Power Set (another axiom), and mappings (trivially constructed via other ZFC axioms). Given that axioms are not subject to disproof (not even in doronetics, we hope), you should focus your efforts on the constructions.

Good luck.

Already done in http://www.internationalskeptics.com/forums/showpost.php?p=10013503&postcount=3712.
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No, that would be a post where you simply reiterated bare assertions. And you paid no respect whatsoever to the required ZFC context.
 
EDIT:

jsfisher said:
At the set-theoretic level, cardinality is a relative measure.
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Please explain it in more details (do you mean that something like Hilbert's paradox of the Grand Hotel is used? (in that case please see http://www.internationalskeptics.com/forums/showpost.php?p=10015020&postcount=3717)).


jsfisher said:
Cantor's Theorem relies on the Axiom of Infinity (an axiom),... Given that axioms are not subject to disproof (not even in doronetics, we hope), you should focus your efforts on the constructions.
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This is exactly what I did, by distinguish between the level of being a set, and the level of being a member of a given set.

You simply ignore the following (so this time please do not do that, if you really wish to discuss about what you call doronetics):

|{}|=0 , |{{}}|=1 |{{},{{}}}|=2 etc. , or |{{}, {{}}, {{},{{}}}, ... }| < aleph0 exactly because the level members is inaccessible to the level of being a set (where the abstract notion of being a set (that is not understood by your naive reasoning about sets, and therefore called by you "Gibberish") is notated by the outer "{" and "}").

In other words, ZFC Axiom of Infinity is wrongly used by Cantor in order to establish his Transfinite universe, exactly because he does not understand that the level of members is inaccessible to the level of being a set (as demonstrated above).

As about infinity, your naive reasoning about sets unable to distinguish between potential infinity (which is at the level of members) and actual infinity (which is at the level of being a set (and notated here by the outer "{" and "}")).

Since |N| (which is at the level of members) < aleph0 (which is at the level of being a set), Cantor's theorem (which according to its reasoning |N|=aleph0) does not hold.
 
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As for Hilbert's paradox of the Grand Hotel, in http://worldsciencefestival.com/blog/first_evidence_for_primordial_ripples_in_space Prof. Brian Greene says:

"The results also affirm, once again, the astounding power of mathematical analysis to lead the way into the most remote corners of creation".

I wish to share with you (once again) some of my mathematical insights, which are associated with the concept of infinity, as related to multiverse point of view.

It is well known that the notion of sets is one of the fundamentals of many mathematical developments. For example {1, 2, 3, 4, 5, …} can be considered as the universe of all natural numbers.

Multiset is a generalization of the concept of sets such that {a, a} ≠ {a}. According to this point of view {1, 2, 3, 4, 5, …} is some form of universe where, for example, {{1, 2, 3, 4, 5, …}, {1, 2, 3, 4, 5, …}} is some form of multiverse.

By using comparisons among objects of two universes under a given multiverse, one enables to get results that are impossible from a single universe point of view, and in order to show it, let us observe Cantor's Transfinite universe from the multiverse point of view.

Unlike in the case of a single universe (where the cardinality of an infinite set is inseparable from its identity), in the case of multiverse it is possible to show different cardinalities that are related to sets that have exactly the same objects.

In order to show it, let us define couple of different mappings between the members of two universes of natural numbers under a given multiset (which is equivalent here to the concept of multiverse), such that each compared universe is the set of all natural numbers:

This comparison is equivalent to the comparison as done in a single universe:
Code: 
{
 {1,2,3,4,...},
  ↕ ↕ ↕ ↕  
 {1,2,3,4,...}
}

These comparisons can be done only under a given multiverse (or multiset, if mathematical terminology is used):
Code: 
{
 {1,2,3,4,...},
  ↓ ↕ ↕ ↕  
   {1,2,3,...}
}

Code: 
{
 {2,4,6,8,...},
  ↓ ↕ ↕ ↕  
   {1,2,3,...}
}

Please pay attention that the comparison under a given multiverse has more degrees of freedom, which enable to use a mapping (a function) that has an input but it does not return any output. As a result the two compared universes do not have the same cardinality even if they have exactly the same members (the same identity).

Moreover, it is also shown that the notion of proper subset is different in the case of multiverse. For example the cardinality of a universe of all even numbers can be greater than the cardinality of a universe that have the all natural numbers (as shown in the last example above).

My point is this: if multiverse is used as a fundamental concept of our framework, whether it is physical or abstract (in the mathematical sense) I think that we have to re-consider fundamental notions that are related to such framework.

As much as I know, such re-consideration was not done yet by mathematicians that develop the foundations of Mathematics, and in my view this re-consideration has to be done also by any framework which deals with infinity.
 
doronshadmi said:
No to your no.

You are invited to abstractly show exactly how the outer "{" and "}" or the void between {} expression, are in the scope of (abstract or non-abstract) existing members, if you really wish to support your "no".
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doronshadmi said:
For example, {} is a mathematical expression of the, so called, empty set.

The cardinality of a given set is defined by the number of its members, so in the case of {} the cardinality is |{}| = 0.

0, in this case, means that the outer "{" and "}" or the void between {} expression, are not in the scope of (abstract or non-abstract) existing members (more precisely, "{" and "}" is "above" existing members, and the void between {} expression is "below" existing members).
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So let's start with what you got right. The cardinality of the empty set is zero.

Let's now talk about what you got wrong.

  1. It is called the empty setWP. A set that has no members is empty, hence the empty set. It is not "the so called 'empty set'".
  2. You believe that the { and the } are parts of a set. They are not. Just like the . is not a word but it is used in grammar / sentence construction to show the end of a sentence.
  3. There is no "above" or "below". Either something is in a set or it's not.
  4. The rest of the quoted post is in error because you either don't understand set construction or set notation. My guess is both.
 
Little 10 Toes said:
Doronshadmi, why can't you start comparing two sets at their first element? Some of your problems will disappear when you do.
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Little 10 Toes, by using the concept of mathematical multiverse, one can choose to start comparing two sets at their first element, or not (and I clearly demonstrate these two options in http://www.internationalskeptics.com/forums/showpost.php?p=10015020&postcount=3717).

This is not the case with mathematical universe (that does not have the degrees of freedom of comparison of a given multiverse).

Your problems about the considered subject will disappear when you distinguish between multiverse and universe.
 
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