
Re: Nathan counts the powerset
Posted:
Dec 28, 1998 2:53 PM


In article <36873940.25248175@news.vsat.net>, mad@ashland.baysat.net (Nathaniel Deeth) wrote: > On Thu, 24 Dec 1998 20:14:01 GMT, ullrich@math.okstate.edu wrote: > > >In article <368253D9.30DF6D70@ashland.baysat.net>, > > Mike Deeth <mad@ashland.baysat.net> wrote: > >> > >[...] > >> Here are ten easy questions requiring yes/no answers. I think I'll learn alot > >from your > >> answers. > >> > >> Part I Questions: (see table below) > >> =================================== > >> (1) Is there a bijection between the natural numbers and the set of natural > >numbers? > > > > If it has to be yes or no the answer is no. But that's like asking > >yes or no, have you stopped beating your wife, or yes or no, have you > >stopped posting stupid messages on sci.math. A much better answer is to > >say the question makes no sense. Here's why: > > > > We talk about a bijection between two _sets_, not a bijection > >between two tomatoes or a bijection between a number and a porcupine. > >When people talk about "a bijection between the natural numbers > >and..." they really mean "a bijection between the set of natural > >numebrs and..."  we all (usually) understand that's what's meant. > > So the question "(1) Is there a bijection between the natural > >numbers and the set of natural numbers?" doesn't _really_ make any > >sense, because you're not asking about a bijection between two > >sets. People talk this way all the time, but if we take your > >question and use the standard translation into precise language > >the question becomes > > > >"(1') Is there a bijection between the set of natural numbers and the > >set of natural numbers?" > > > >Now, the answer to _that_ question is obviously yes. But that > >question is also obviously not what you really meant to ask, > >so it doesn't seem too relevant. > > > > >"(1') Is there a bijection between the set of natural numbers and the set of natural numbers?" > > > >Now, the answer to _that_ question is obviously yes. But that question is also obviously not what you really meant to ask, >so it doesn't seem too relevant. > > > > No! That *is* the question I intended. And it *is* extreemly > relevant. So let me repeat the answer: "there is a bijection between > a set of ALL natural numbers and a set of ALL natural numbers." > > Am I going too fast? Have I lost you? Are my ideas too complicated? > ;)
You should _really_ get your facts straight before making such an ass of yourself over and over. You insist you _really_ meant to be asking
"Is there a bijection between the set of natural numbers and the set of natural numbers?"
??? That's an extremely stupid question (I'm still certain it's not the question you intended.) The _identity_ function, defined by the formula f(n) = n, is a bijection between the set of natural numbers and the set of natural numbers.
Saying
"there is a bijection between a set of ALL natural numbers and a set of ALL natural numbers."
sounds very very strange, because of the "a". There's only one "set of all natural numbers".
> Is there just one set of natural numbers, or can there be more? ie. A > complete set of *red* natural numbers and a complete set of *blue* > natural numbers? To answer this question we need to examine Peano's 5 > axioms. > > 1) There is a first Natural Number. > 2) Every Natural Number has a successor. (This is what counting is > about) > 3) The First Natural Number is not the successor of any other > Natural Number. (Counting does not go in circles.) > 4) Two natural numbers which have the same successor, ARE the same. > (No branching.) > 5) If you have a set of Natural Numbers, including the first number, > and including the successor of every number already in the set, then > you have ALL the natural numbers. (Axiom of Induction.)
Actually these are not Peano's axioms. The axioms don't say anything about "Natural Numbers" per se, they're purely logical axioms. Which happen to be true of the natural numbers, as well as of various other constructs.
> Notice that none of the axioms prohibit the existance of more than one > set of ALL Natural Numbers.
Nor does this fact imply there _is_ more than one set of all Natural Numbers.
> Below are some examples of Natural Numbering systems conforming to > Peano's 5 axioms. Each system looks differant but, in fact, they are > isomorphic (mathematically identical).
You don't need to keep explaining what the word "isomorphic" means. Yes, these systems are all isomorphic to N. SO WHAT? The existence of several things isomorphic to N does not imply that N itself IS more than one thing.
> > Is the Powerset of ALL the Natural Numbers countable? In other words, > can Peano's Naturals be put in bijection with the Powerset of Peano's > Naturals? In fact, the Powerset of ALL the Natural Numbers satisfies > all 5 of Peano's axioms. Therefor the Powerset of ALL Natural Numbers > *is* the set of ALL Natural Numbers. (aka Peano's naturals) It is > also known (obviously) that the set of ALL Natural Numbers can be in > bijection with the set of ALL Natural Numbers. Therefor, the the > Powerset of Peano's Naturals is denumerable (can be put into bijection > with the Natural Numbers). Could anything be simpler than that? :)
This is sheer nonsense. The powerset satisfies Peano's axioms? It's nonsense formally, it's also false. It's meaningless gibberish formally: a _set_ cannot _possibly_ satisfy those axioms, what may orr may not satisfy the axioms is a set together with a "successor" function defined on that set. To make sense of what you said you need to say
"the power set of N satisfies Peano's Axioms, where we define the successor of a set A by the formula S(A) = ____________"
How _do_ you define the "successor" of a set of natural numbers?
> Nathaniel Deeth > Age 11 > >
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