Re: Non-zero gaps between real numbers

On Dec 1, 10:44 pm, Marshall <marshall.spi...@xxxxxxxxx> wrote:
On Dec 1, 9:24 am, Venkat Reddy <vred...@xxxxxxxxx> wrote:

On Dec 1, 9:47 pm, Marshall <marshall.spi...@xxxxxxxxx> wrote:
On Dec 1, 5:11 am, Venkat Reddy <vred...@xxxxxxxxx> wrote:

Assume a sentence.
Prove a contradiction based on that sentence.
This proves the negation of the assumed sentence.

What is your understanding of RAA? Please be specific.

I've not questioned your proof's validity. It just doesn't address the
given problem. Read below.

Tisk, tisk, you didn't answer my question. My question was,
what is your understanding of RAA? Please be specific.

That is not pertinent. Your proof using RAA is fine for showing that
there is always a new number betwen two given number. However the
problem was something different: to show that this new number can
"fill" the measure between the given numbers.


But you haven't made any progres on the problem
posed by my second sentence. You haven't proved that there is no non-
zero gap.

I have shown that if there is *any* gap, then a contradiction
results.

No. You have just shown that if there is *any* gap, then it is
possible to find a new number in it.

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. When you add a new number into
existing gap, you just divide it, but not really filling it.

So what I have shown is that if there are two real numbers
with no numbers between them then those two numbers have
a number between them. Do you see why that is a contradiction?

In other words, if there is a gap then it is not a gap. That
means there cannot possibly be any gap.

But it doesn't pursue the new
gaps until they are diminished to zero.

What new gap? There cannot be any gap. Anything that
we might assume to be a gap has numbers in it, hence
is not a gap. Any possible "new" gap, any possible "old"
gap, anything that could potentially be considered to be
a gap of any kind, has numbers in it and hence is not a gap.

So you are sure that the gaps will always have a number (mid-point)in
them even after finding infinite number of such midpoints. But have we
reached zero yet on the gap size? Definitely no. Do you think we can
still continue beyond infinity, until it hits zero? If not, then we
are left with non-zero gaps.

All these questions are moot, since there are no gaps. Gaps
are an impossibility, so we don't have to consider any such
questions. We don't "reach" anything. We don't "continue".
We're not engaged in some process. Gaps are an impossibility,
so any "process" we might have in mind that involves
gaps can never even begin.

So, here is the summary:

You have proved that it is posisble to find a new real number that
lies between any two given real numbers.

This new number can only divide the existing gap into two parts, but
not fill the existing measure between the given numbers. So, to proove
that gaps are zero, you will have to apply the results of your prof
iteratively until the gap gets smaller and smaller and diminish to
zero, which might not even happen at infinite number of such
divisions.

So your attempt to prove that there are no gaps not only involved
infinity but also seen failing at infinity.

- venkat
.

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