Re: Non-zero gaps between real numbers

"Marshall" <marshall.spight@xxxxxxxxx> wrote in message
news:dc67d4fb-763a-4314-b043-8fee1813ffbf@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
On Dec 1, 10:24 am, Venkat Reddy <vred...@xxxxxxxxx> wrote:
On Dec 1, 10:44 pm, Marshall <marshall.spi...@xxxxxxxxx> wrote:

What is a gap? A gap is two real numbers with no number between
them.

No. Gap is being unable to completely fill the measure between two
given numbers with new numbers.

If there are two real numbers that aren't "completely fill[ed]" in
between them, then somewhere inside there are two real
numbers that have *no* real numbers between them.

This is not obvious and would require proof...

Which my modest proof shows to be impossible.

Can you state precisely what you mean by "completely
fill[ed]?" Give me a formula that tells whether a pair
of numbers constitute a filled gap, or whatever the
terminology is you'd like to use. If you can't state it,
then you can't talk about what properties it has.

The formula I came up with was

EaEb!Ez(a<z<b)

where the domain of discourse is real numbers.
That to me defines a gap. There is an a and a
b such that there is no z that is between them.
No infinities. Just existence and nonexistence.

Why does there have to be such an a and b? E.g. consider the set

{x in R: x < -1 or x > +1}

There is clearly a gap in this set, but there is no a and b matching your
definition...

Regards,
Mike.


Your definitions. Show them to us.


Marshall


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