July 17th, 2017, 09:31 AM  #31 
Member Joined: Oct 2009 Posts: 88 Thanks: 28  
July 17th, 2017, 01:46 PM  #32 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,079 Thanks: 87  
July 17th, 2017, 01:54 PM  #33 
Member Joined: Oct 2009 Posts: 88 Thanks: 28  OK, please give me that definition. Then prove me that every decimal representation corresponds to a natural number in a unique way and vice versa. Come on, don't just state things. Actually prove them too! I'd prefer if you'd rely on the Peano axioms alone, but hey, it's up to you. 
July 17th, 2017, 02:39 PM  #34  
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,079 Thanks: 87  Quote:
Quote:
If any a$\displaystyle _{n}$'s are different, the N's are different. If you can conceive of an arbitrary small $\displaystyle \delta$, then you should be able to conceive of an arbitrarily large 1/$\displaystyle \delta$. What is pi? 3.141592654......, by definition. It's the unique infinite sequence(s) that define pi. EDIT: If N=$\displaystyle \sum_{n=1}^{\infty}a_{n}\times 10^{n}$ then N+1 is of the same form. If N=$\displaystyle \sum_{n=1}^{\infty}a_{n}\times 10^{n}$ is a natural number, then so is it for n=n+1. By elementary arithmetic. Last edited by zylo; July 17th, 2017 at 03:05 PM.  
July 17th, 2017, 03:12 PM  #35 
Member Joined: Oct 2009 Posts: 88 Thanks: 28  That's a rather bad definition since it doesn't define pi uniquely. The problem is that we don't know all the digits of pi, so we shouldn't define pi by its decimal expansion. Not saying that pi doesn't have a decimal expansion, it's just not possible to define pi this way. So perhaps let us start this way. What is your definition of the natural numbers and of the real numbers? Last edited by skipjack; July 17th, 2017 at 03:17 PM. 
July 17th, 2017, 03:18 PM  #36 
Global Moderator Joined: Dec 2006 Posts: 17,466 Thanks: 1312 
As you, zylo, referred to a definition to support your assertion, the onus is on you to provide that definition, not Micrm@ss. I don't know of any accepted definition that supports your assertion that any infinite binary sequence defines a unique natural number. Your mention of pi isn't relevant in this matter, as pi isn't a natural number (it's a real number, though).

July 17th, 2017, 05:27 PM  #37  
Senior Member Joined: Jun 2014 From: USA Posts: 299 Thanks: 21  Quote:
Naturally, there will be an infinite number of binary sequences for each natural number. There are infinitely many surjective mappings from the set of infinite binary sequences onto the natural numbers. Instead of trying to contradict Cantor's Diagonal Argument (which is still your underlying focus here...), I challenge you to try and do the opposite. Try to prove to yourself that it is true. And by that, I do not simply mean for you to try and believe in the theorem, I mean for you to try and prove to your own satisfaction that it is true. In the end, you will never disprove Cantor's Theorem. It's not possible to disprove. Your best shot is to wage war on any use of the axiom of infinity. Set theory's current state of affairs is one of "assume a certain axiom or conjecture, and then X is (not) provable, but don't, and then Y is (not) provable." The continuum hypothesis is one example. Given the standard axioms, we can neither prove nor disprove it. We can conjecture the hypothesis is true and derive certain results, but we can likewise conjecture it is false and derive other results. In a sense, it isn't math any more. Hugh Woodin makes a similar comment in his lecture here (he is an extremely respected set theorist). My personal intuition is that the axiom of infinity, once accepted, will always lead to uncertainty. Compare that intuition with Gödel's incompleteness theorems showing that a complete and consistent set of axioms for all of mathematics is not possible. If you want to waste your life slaving away trying to disprove a theorem you simply cannot disprove, then I'm sorry to hear that. Your motivation for doing so cannot be sane, quite frankly. It's like flapping your arms because you're convinced you can fly. But, if you want to try and get onto the forefront of set theory, perhaps there is something more left to prove. Find a resolution to Scott's theorem by showing whether or not a measurable cardinal exists (ie, if there exists a measurable cardinal, then $V \neq L$). Perhaps that would net you a Nobel Prize. In the very least, your name would go down in history (that appears to be your delusional goal with respect to disproving Cantor's Theorem...).  
July 17th, 2017, 07:16 PM  #38 
Math Team Joined: Dec 2013 From: Colombia Posts: 6,854 Thanks: 2228 Math Focus: Mainly analysis and algebra  
July 17th, 2017, 07:18 PM  #39 
Math Team Joined: Dec 2013 From: Colombia Posts: 6,854 Thanks: 2228 Math Focus: Mainly analysis and algebra  
July 17th, 2017, 07:19 PM  #40  
Senior Member Joined: Jun 2014 From: USA Posts: 299 Thanks: 21  Quote:
There have been a number of prizes awarded to mathematicians, but these were for their contributions to Physics. Wooden also discusses set theory's potential application to Physics in the above lecture, but I have my doubts... http://wwwhistory.mcs.standrews.ac...urs/Nobel.html Last edited by AplanisTophet; July 17th, 2017 at 07:22 PM.  

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