Re: Uncomputable numbers are all in your head

Scott wrote:
Let me ask a question that is a bit tangential but still related. I
have read on the posts people claiming that a denumerable infinite
binary string cannot be mapped to an integer because integers are not
infinite. Yet I cannot find any reference to this in the textbooks.
Can you provide a bit of background of why this is so?

Your terminology is not standard. So, I'm not sure what you mean or what
you think.

The correct theorem is that the set of infinite binary strings is
uncountable, i.e., there is no bijection from N to the set of infinite
binary strings.

You will never find a math book that includes the phrase "because
integers are not infinite". You only find such statements in reply to
nonsensical posts that try to establish a bijection by claiming
(erroneously) that an infinite string is a natural number. The standard
notation for natural numbers uses finite strings of digits. Anyone
claiming that an infinite string represents a natural number must
explain what natural number the string represents.

--
David Marcus
.

Relevant Pages

  • Re: infinity
    ... >>> Why are you assuming that there is a longest word in the language? ... >> finite, that means none are infinite, therefore S^L is not infinite either. ... > I am not assuming that there is a longest word. ... > the maximum string length, i.e. the largest finite natural number. ...
    (sci.math)
  • Re: .99999... still=/= 1
    ... >>2) The point of reference means how we define the counting results ... the string length and the members in the placeholders of the string. ... Note that supernatural numbers are that I have called earlier infinite ... omega 1 is behind our standard infinity in the next infinity. ...
    (sci.math)
  • Re: Logarithm of transfinite numbers
    ... or the value of the string I am discussing. ... There is no point at which any finite natural has an infinite ... They ARE the finite naturals. ... At which point has my abstraction differed from what you're ...
    (sci.math)
  • Re: Logarithm of transfinite numbers
    ... Tony Orlow wrote: ... Can all 1's up to any finite position represent an infinite ... There is no infinite position in the string. ... And no sum ending at M represents the sum over the ...
    (sci.math)
  • Re: Thinking about TOs theories about Infinity
    ... each succeeding string of 1/2 the previous height, ... bitstring, being a copy of the bitstring from the previous ... Tony Orlow wrote: ... result in an infinite string. ...
    (sci.math)