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October 21st, 2018, 05:39 PM   #231
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 Originally Posted by Pengkuan I have shown in my article ...
If the majority is always wrong, what happens if the majority holds that opinion? October 21st, 2018, 07:19 PM #232 Senior Member   Joined: Oct 2009 Posts: 805 Thanks: 306 So the majority is always wrong? Got it? The majority seems to believe in 1+1=2. So the majority is wrong and 1+1=1 instead. In fact, accepting that 1+1=2 causes a lot of problems that we can't solve! If we'd accept that 1+1=1 instead then this would mean that every number equals 1, and hence every problem is solved. Pengkuan, this is what your latest post sounds like to people who know math. October 21st, 2018, 07:44 PM   #233
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 Originally Posted by Maschke If the majority is always wrong, what happens if the majority holds that opinion?
The majority is not always wrong. In very large majority of case, the majority is right. This is why the world runs so smoothly. The majority is wrong only in rare particular cases, when breaking wall of thinking. This is why the great breakthroughs in history are all done by lonely individuals like Newton, Copernicus or Einstein.

For set theory, the majority does not see set theory as blocaded by a wall. They see continuum hypothesis and Cantor's Paradox as isolated problems that can be solved by adding some axioms. They are happy with the present theory and does not search for change. But if the blocking is from the deep , the foundation, adding some axioms to old theorems will not break the wall.

Secondly, the majority believe in old theorems. They will not be convinced by only one person, but they need the endorsement from everyone. This is why they stay within the wall and do nothing to evade from the small comfort zone and they reject new non orthodoxical ideas as evidently wrong.

So, breakthroughs are never done by group of people and in consequence, breakthrough in set theory will not come from the majority .

What happens if the majority holds that opinion? The science stagnates. October 21st, 2018, 07:46 PM   #234
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 Originally Posted by Micrm@ss So the majority is always wrong? Got it? The majority seems to believe in 1+1=2. So the majority is wrong and 1+1=1 instead. In fact, accepting that 1+1=2 causes a lot of problems that we can't solve! If we'd accept that 1+1=1 instead then this would mean that every number equals 1, and hence every problem is solved. Pengkuan, this is what your latest post sounds like to people who know math.
Repeat:
The majority is not always wrong. In very large majority of case, the majority is right. This is why the world runs so smoothly. October 21st, 2018, 08:27 PM   #235
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 Originally Posted by Pengkuan That is, the set of all sets cannot have a cardinal number, or cannot exist. If instead, the cardinal number of the power set of a set X equals the cardinal number of the set X,
The set of all sets cannot exist because it would have to contain itself. The fact that there is no greatest infinity is a feature of the system, not a bug.

We know that the cardinalty of the power set is greater than the cardinality of the set for all finite sets. Why should we insist, without evidence, that this change for infinite sets?

You can't "reject" Cantor's theorem, it has been proved to be true. You either change your axioms or find a flaw in all proofs and demonstrate a counter proof. October 22nd, 2018, 05:03 AM   #236
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 Originally Posted by Pengkuan Your question is characteristic of the blocking of set theory. In order to unblock, can we first think of the foundation of set theory without prejudice such as, Cantor’s theorem is correct or my proposition is incorrect...
Mathematical theorems are truthful statements devoid of prejudice. You're asking for us to believe you over a proof to the contrary. You have a better chance of convincing me of the (non)existence of God.

Quote:
 Originally Posted by Pengkuan ...see how the notion of multiple cardinal numbers create obstacle in set theory. The continuum hypothesis does not find solution within ZFC set theory, not because we are unable to find it, but because it does not exist. In fact, not only we have “continuum difficulty” of first degree, that is, between $\aleph_0$ and $\aleph_1$, but also between $\aleph_1$ and $\aleph_2$ and all degrees above.
If it doesn't exist, then we wouldn't be able to find it...

Anyways, CH may be independent of ZFC, but that doesn't mean a solution to CH doesn't exist. There may be some intuitive axiom out there yet to be discovered that people readily accept and that changes things.

Quote:
 Originally Posted by Pengkuan Another problem is Cantor's Paradox: There is no greatest cardinal number. That is, the set of all sets cannot have a cardinal number, or cannot exist. This paradox is created by Cantor’s theorem because power set creates ever higher cardinal number with no end, but the set of all sets is the end.
The set of all sets cannot exist. See Russell's paradox:

Where there is no set of all sets, I have no idea why you would think there should be a set having a cardinality greater than that of all other sets.

Quote:
 Originally Posted by Pengkuan If instead, the cardinal number of the power set of a set X equals the cardinal number of the set X, that is, power set does not increase cardinal number, what will become the continuum hypothesis and Cantor's Paradox? Actually, these problems will vanish by themselves. For example, if the power set of a set X equals the cardinal number of the set X, then all set will have the same cardinal number and thus, the set of all sets can exist and will have the cardinal number $|\mathbb{N}|$.
If "the cardinal number of the power set of a set $X$ equals the cardinal number of the set $X$," then we'd have weird crap like...:

$$X = \{1,2\} \implies |X| = 2 \implies |\mathcal{P}(X)| = 2 \neq 4$$

...which breaks my heart because you just agreed that for any set $X$, the cardinality of the power set of $X$ equals $2^{|X|}$.

Further, your notion that all infinite sets may have the same cardinality doesn't solve Russell's paradox, so the set of all sets could not exist as you implicate.

Quote:
 Originally Posted by Pengkuan For the continuum hypothesis, if the power set of $\mathbb{N}$ equals the cardinal number of $\mathbb{N}$, then, the cardinal number of all sets is $\aleph_0$. So, the cardinal number of the real numbers is $\aleph_0$ too. Then, the continuum hypothesis for the sets of real numbers and for all the sets of higher degree disappears.
That would be true if $|\mathbb{N}| = |\mathcal{P}(\mathbb{N})|$, but it doesn't. Alternatively, to avoid the continuum hypothesis, you could remove the axiom of power set from ZF set theory.

Quote:
 Originally Posted by Pengkuan Above is the consequence of a hypothetical rejection of Cantor’s theorem, which will make set theory much coherent, which will also solve a philosophical problem, that of the uniqueness of infinity. It is really paradoxical that there are infinities above infinity. Philosophically, infinity is the absolute great, above which nothing should exist.
It would also make set theory inconsistent unless your method of rejecting Cantor's theorem was to discard axioms that allow for it, such as the axiom of infinity.

You're letting your personal feelings get the best of you in that you want mathematical proofs to follow your belief that "infinity is the absolute great, above which nothing should exist." Despite having a greater cardinality, I dare say that $\mathcal{P}(\mathbb{N})$ is not a greater infinity than $\mathbb{N}$, nor is it 'larger', more awesome, etc. The term 'cardinality' has a defined mathematical meaning that is not plagued by subjective interpretation.

Quote:
 Originally Posted by Pengkuan Now that set theory is hitting a wall, why do we insist in doing the same thing by keeping Cantor’s theorem? Why would not we do new things beyond Cantor’s theorem? Maybe it is Cantor’s theorem that creates the obstacle and the wall. We have nothing to lose by trying new ways of thinking
You really missed the whole notion of consistency, didn't you? I will grant you that there may be something to learn from studying inconsistent systems. The foundations of mathematics have been painfully crafted to eliminate inconsistency however, so you can't simply ignore Cantor's theorems and throw them out. You would have to find an error in the proofs in order to show that adhering to them would be inconsistent before you could create your personal utopia where sets all come in one infinite size.

Quote:
 Originally Posted by Pengkuan I have shown in my article « Analysis of the proof of Cantor's theorem » the flaw of Cantor’s theorem No buddy, you haven't. You haven't even come close, and since you're going to act ridiculous, I'll accept your invitation to have a laugh at your expense.

Quote:
 Originally Posted by Pengkuan I have shown in my article ...in «Building set and counting set» how to count the power set of $\mathbb{N}$. Face is starting to hurt now... You can't count it because it's not countable and all you've shown is how to enumerate the finite elements of $\mathcal{P}(\mathbb{N})$. I thought we covered this.

Quote:
 Originally Posted by Pengkuan So, I do not agree that $|\mathbb{N}| = \aleph_0 < |\mathcal{P}(\mathbb{N})| = 2^{\aleph_0}$. In the contrary, I think it is the problem itself.
You can't disagree either because you have no proof to the contrary.

I guess that's as far as we can get on this topic. Go find your proof to the contrary and, until then, mathematicians will keep accepting Cantor's proof.

Last edited by AplanisTophet; October 22nd, 2018 at 05:11 AM. October 22nd, 2018, 05:10 AM   #237
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 Originally Posted by AplanisTophet Mathematical theorems are truthful statements devoid of prejudice. You're asking for us to believe you over a proof to the contrary. You have a better chance of convincing me of the (non)existence of God. If it doesn't exist, then we wouldn't be able to find it... Anyways, CH may be independent of ZFC, but that doesn't mean a solution to CH doesn't exist. There may be some intuitive axiom out there yet to be discovered that people readily accept and that changes things. The set of all sets cannot exist. See Russell's paradox: https://plato.stanford.edu/entries/russell-paradox/ Where there is no set of all sets, I have no idea why you would think there should be a set having a cardinality greater than that of all other sets. If "the cardinal number of the power set of a set $X$ equals the cardinal number of the set $X$," then we'd have weird crap like...: $$X = \{1,2\} \implies |X| = 2 \implies |\{ \mathcal{P}(X)| = 2 \neq 4$$ ...which breaks my heart because you just agreed that for any set $X$, the cardinality of the power set of $X$ equals $2^{|X|}$. Further, your notion that all infinite sets may have the same cardinality doesn't solve Russell's paradox, so the set of all sets could not exist as you implicate. That would be true if $|\mathbb{N}| = |\mathcal{P}(\mathbb{N})|$, but it doesn't. Alternatively, to avoid the continuum hypothesis, you could remove the axiom of power set from ZF set theory. It would also make set theory inconsistent unless your method of rejecting Cantor's theorem was to discard axioms that allow for it, such as the axiom of infinity. You're letting your personal feelings get the best of you in that you want mathematical proofs to follow your belief that "infinity is the absolute great, above which nothing should exist." Despite having a greater cardinality, I dare say that $\mathcal{P}(\mathbb{N})$ is not a greater infinity than $\mathbb{N}$, nor is it 'larger', more awesome, etc. The term 'cardinality' has a defined mathematical meaning that is not plagued by subjective interpretation. You really missed the whole notion of consistency, didn't you? I will grant you that there may be something to learn from studying inconsistent systems. The foundations of mathematics have been painfully crafted to eliminate inconsistency however, so you can't simply ignore Cantor's theorems and throw them out. You would have to find an error in the proofs in order to show that adhering to them would be inconsistent before you could create your personal utopia where sets all come in one infinite size. No buddy, you haven't. You haven't even come close, and since you're going to act ridiculous, I'll accept your invitation to have a laugh at your expense. Face is starting to hurt now... You can't count it because it's not countable and all you've shown is how to enumerate the finite elements of $\mathcal{P}(\mathbb{N})$. I thought we covered this. You can't disagree either because you have no proof to the contrary. I guess that's as far as we can get on this topic. Go find your proof to the contrary and, until then, mathematicians will keep accepting Cantor's proof.
The sad thing about people who reject Cantor, is that they could actually very very easily prove their point to all mathematicians. You just find a completely formalized proof of Cantor's theorem (many are online), and you just find the step which doesn't follow from either a simple axiom, or an inference rule.
Just giving this very step is enough to generate a crisis among mathematicians. They will be forced to believe you, and a fields medal is almost certain.

But Cantor disbelievers don't do this. They just keep giving vague or metaphysical arguments. They misrepresent definitions, and keep making basic errors. Nobody will ever believe something like this. Ever. October 22nd, 2018, 03:41 PM   #238
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 Originally Posted by v8archie The set of all sets cannot exist because it would have to contain itself. The fact that there is no greatest infinity is a feature of the system, not a bug. We know that the cardinalty of the power set is greater than the cardinality of the set for all finite sets. Why should we insist, without evidence, that this change for infinite sets? You can't "reject" Cantor's theorem, it has been proved to be true. You either change your axioms or find a flaw in all proofs and demonstrate a counter proof.
Thanks to all. October 22nd, 2018, 03:59 PM   #239
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 Originally Posted by v8archie The set of all sets cannot exist because it would have to contain itself.
You know, that's a common misconception. It's not true.

* First, it's perfectly ok for a set to contain itself. The only reason a set can't contain itself is that we make an axiom to that effect, the axiom of regularity. It's also known as the axiom of foundation.

Absent regularity, you get non well-founded set theory. Yes it's a thing! An obscure thing but a thing nonetheless.

So sets can contain themselves; and that in itself does NOT preclude there being a set of all sets.

What disallows the set of all sets is unrestricted set comprehension. Frege thought a set could be characterized by a property, or predicate. Such as the set of all apples, the set of all galaxies, the set of all real numbers. But if your predicate is $x \notin x$, you immediately get a contradiction. So you can NOT necessarily define a set by a predicate.

The solution is restricted comprehension, expressed as the axiom schema of specification. It's an axiom schema because it's actually a factory for producing axioms. Namely, given a predicate P, we make an axiom that says: For any set X, there is a subset of X such that P is true exactly on the members of that subset.

The difference is that before we apply a predicate, we must first have a set. Then Russell's paradox goes away.

tl;dr: The way we resolve Russell's paradox is by outlawing unrestricted comprehension, and instead adopting the axiom schema of specification. It has nothing to do with sets containing themselves. That's just the predicate we use to get the contradiction. It's possible for sets to contain themselves unless you make an axiom that says they can't.

Last edited by Maschke; October 22nd, 2018 at 04:02 PM. October 22nd, 2018, 04:02 PM   #240
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Quote:
 Originally Posted by Maschke You know, that's a common misconception. It's not true.
If you change your axioms, you can obviously make almost anything happen.  Tags binaryexpressed, cardinality, continuum hypothesis, diagonal argument, numbers, real, set Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post topsquark Abstract Algebra 54 August 17th, 2015 07:46 AM Tau Applied Math 4 January 18th, 2014 12:40 PM mente oscura Number Theory 7 June 1st, 2013 02:48 AM farleyknight Applied Math 3 December 20th, 2008 08:01 PM johnny Computer Science 6 October 18th, 2007 10:29 AM

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