Re: how to list all of the real numbers

In article <1186702847.280483.145990@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
lwalke3@xxxxxxxxx wrote:

On Aug 9, 9:24 am, "Ross A. Finlayson" <r...@xxxxxxxxxxxxxxx> wrote:
Denying the Axiom of Infinity doesn't necessarily make on a finitist:
instead perhaps there is some consideration that the characterization
of an infinite set by the axiom of infinity as generally stated (that


I see what you mean. In other words, what you are suggesting is that
one replace the Axiom of Infinity with another axiom phi, such that
ZF-Infinity+phi proves the existence of a set that is not hereditarily
finite, but one can prove certain theorems that match one's intuition
yet whose negations are provable in ZFC.

No, not everyone who thinks Cantor is not the end-all be-all when it
comes to infinity is a crank.

But standard mathematicians, of course, believe otherwise.

Standard mathematicians do not tend to call a person a crank for merely
disagreeing with Cantor's results , but standard mathematicians tend to
think of those who call Cantor a crank as being themselves cranks.

If a function bijects between the natural integers and reals then it
shares a variety of properties with the Equivalency Function, which
maps the first integer after zero to the first real number after zero
in the total linear and well-ordering there of the reals, the natural
order. That the value EF(1) would be not only an infinitesimal,
hinting that the real numbers contain infinitesimals, but furthermore
a value nearer zero than any other value, contradicting the density of
the reals within themselves as always having infinitely many between
any two, leads to some non-standard notions of the real numbers.

I agree that infinitesimals are possible, but unfortunately most
mathematicians will only consider standard theories.

Ever since Robinson, infinitesimals have been accepted as possible.
.

Relevant Pages

  • Re: Cantors "diagonal argument". My Objection.
    ... Infinity is NOT relevant to the proof ... you DENY the axiom of infinity (if you replace it with an axiom ... It DOESN'T MATTER whether you restrict these reals to being between 0 ... This IS NOT specific to Cantor IN ANY way except that he got there ...
    (sci.logic)
  • Re: Is continuum completely filled up?
    ... compactification of the reals, as used in analysis. ... the surreals do not satisfy the least upper bound ... If the axiom of power set implys uncountable reals, ... The axiom of infinity guarantees that an infinite set exists. ...
    (sci.math)
  • Re: how to list all of the real numbers
    ... those with the undecideable continuum hypothesis, yet none of them, ... continuum's cardinality is thus simply equivalent to a lesser cardinal ... does not hold where the reals are a set. ... Infinity in numbers does actually exist. ...
    (sci.math)
  • Re: how to list all of the real numbers
    ... does not hold where the reals are a set. ... could be used as a proof for the existence of limit ordinals. ... concept of "adjacent points" is provably inconsistent in geometry. ... and the infinity of the codomain. ...
    (sci.math)
  • diagonal argument on ordered array of reals
    ... 'Construct the reals in combinatorially by increasing ... The question is, as n -> infinity, does this flatten out the array. ... inf = 2 to the inf, the array is square, and we can ...
    (sci.math)
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