Maurice Tutor

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About Maurice Tutor

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Algebra,Applied Sciences,Biology,Calculus,Chemistry,Economics,English,Essay writing,Geography,Geology,Health & Medical,Physics,Science Hide all
Teaching Since: May 2017
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Education

  • MCS,PHD
    Argosy University/ Phoniex University/
    Nov-2005 - Oct-2011

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  • Professor
    Phoniex University
    Oct-2001 - Nov-2016

Category > Management Posted 04 Feb 2018 My Price 10.00

denumerable subset

For any set S, we have |S| < |p="" (s)|.="" the="" function="" g:="" s="" →="" p="" (s)="" given="" by="" g(s)="{s}" is="" clearly="" injective,="" so="" |s|="" ≤="" |p="" (s)|.="" to="" prove="" that="" |s|="" ≠="" |p(s)|,="" we="" show="" that="" no="" function="" from="" s="" to="" p="" (s)="" can="" be="" surjective.="" suppose="" that="" f="" :="" s="" →="" p="" (s).="" then="" for="" each="" x="">∈ S, f (x) is a subset of S. Now for some x in S it may be that x is in the subset f (x) and for others it may not be. Let

 

By applying Theorem 4.18 again and again, we obtain an infinite sequence of transfinite cardinals each larger than the one preceding:

 

Does the cardinal c fit into this sequence? In Exercise 24 we sketch the proof that |p (N) | = c. In Exercise 11 we show that every infinite set has a denumerable subset. Since (by Theorem 4.9) every infinite subset of a denumerable set is denumerable, we see that ℵ0 is the smallest transfinite cardinal. What is the first cardinal greater than ℵ0? We know that c > ℵ0, but is there any cardinal number λ such that

More specifically, is there any subset of R with size “in between” N and R? Experience tells us that there is not, because no such set has ever been found. The conjecture that there is no such set was first made by Cantor and is known as the continuum hypothesis. In 1900 it was included as the first of Hilbert’s famous 23 unsolved problems. Whether it is true or false is still an unanswered—perhaps unanswerable—question. It is known, however, that the assumption of the continuum hypothesis does not contradict any of the usual axioms of set theory. (This was proved by Kurt Gödel in 1938.) But lest we take too much comfort in this, we should also point out that it has been proved (by Paul Cohen in 1963) that the denial of the continuum hypothesis does not lead to any contradictions either. Thus the continuum hypothesis is undecidable on the basis of the currently accepted axioms for set theory. (It can be neither proved nor disproved.) It remains to be seen whether new axioms will be found that will enable future mathematicians finally to settle the issue.

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Maurice Tutor
(5)
Status NEW Posted 04 Feb 2018 07:02 PM My Price 10.00

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