October 21st, 2018, 05:39 PM  #231 
Senior Member Joined: Aug 2012 Posts: 2,209 Thanks: 651  
October 21st, 2018, 07:19 PM  #232 
Senior Member Joined: Oct 2009 Posts: 753 Thanks: 261 
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  
Senior Member Joined: Dec 2015 From: France Posts: 103 Thanks: 1  Quote:
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  
Senior Member Joined: Dec 2015 From: France Posts: 103 Thanks: 1  Quote:
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  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,622 Thanks: 2611 Math Focus: Mainly analysis and algebra  Quote:
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  
Senior Member Joined: Jun 2014 From: USA Posts: 494 Thanks: 36  Quote:
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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:
https://plato.stanford.edu/entries/russellparadox/ 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:
$$X = \{1,2\} \implies X = 2 \implies \mathcal{P}(X) = 2 \neq 4$$ ...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:
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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:
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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:
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:
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  
Senior Member Joined: Oct 2009 Posts: 753 Thanks: 261  Quote:
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  
Senior Member Joined: Dec 2015 From: France Posts: 103 Thanks: 1  Quote:
 
October 22nd, 2018, 03:59 PM  #239  
Senior Member Joined: Aug 2012 Posts: 2,209 Thanks: 651  Quote:
* 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 wellfounded 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 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,622 Thanks: 2611 Math Focus: Mainly analysis and algebra  

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binaryexpressed, cardinality, continuum hypothesis, diagonal argument, numbers, real, set 
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