Re: Non-zero gaps between real numbers

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





On Dec 1, 2:07 pm, Marshall <marshall.spi...@xxxxxxxxx> wrote:
On Nov 30, 8:29 pm, Venkat Reddy <vred...@xxxxxxxxx> wrote:
On Dec 1, 9:13 am, Marshall <marshall.spi...@xxxxxxxxx> wrote:
On Nov 29, 8:01 pm, Marshall <marshall.spi...@xxxxxxxxx> wrote:
On Nov 28, 9:34 pm, Venkat Reddy <vred...@xxxxxxxxx> wrote:

The definition of real numbers allows one to find a real number
between any two given different real numbers. If one uses this to
assert that there is no non-zero extent gap devoid of real numbers in
it, then I think it is not a complete proof but just an assertion.

Do you know what the phrase "reductio ad absurdum" means?
Do you consider it a valid proof technique? Do you see how to
use RAA to prove that there is no non-zero extent gap devoid
of real numbers? If not, would you like me to show the proof?

I am still hoping for an answer to the above.

Yes, I knew what "reductio ad absurdum" means. I don't think you can
prove using RAA and without using any quantity involving infinity.
Once you are using infinity, then you need first proove that RAA is a
valid technique for such qunatities.

Try if you can.

Okay, thanks for the opportunity. I'm definitely a math newbie, so
I might screw this up, but let's see where it goes.

Assume that there exists a non-zero extent gap devoid of real numbers.
That is to say, we have two real numbers, a and b, a < b, with
no real numbers between them. (a and b are different, because a
is strictly less than b, so there's the non-zero extent.)
Consider the number a + (b-a)/2.
This is a real number, because + - are closed over the reals, and / is
closed over the reals provided the second operand is nonzero, and
we're using 2.
a < a + (b-a)/2 < b
So there is a real number between a and b, contradicting our
assumption that there was no such number.
Thus our assumption leads to a contradiction, so by RAA,
the negation of our assumption is true, therefore:
There exists no non-zero extent gap devoid of real numbers.
QED

As requested, no infinity.

You have proved that given two different real numbers we can find a
new real numbers between them. But here is the problem again (same
text from my post):

The definition of real numbers allows one to find a real number
between any two given different real numbers. If one uses this to
assert that there is no non-zero extent gap devoid of real numbers in
it, then I think it is not a complete proof but just an assertion.

So whatever you have proved, I have already taken into account by my
first sentence above.

No that is not correct. If things worked that way, then for any
A that proves B, you could just write some sentences as
follows:

A. If one uses A to assert B, then I think it is not a complete
proof but just an assertion.

Then someone shows you a valid proof of B given A,
and you "no, that proof doesn't work because I already
took that in to account."

It is either a valid proof or it isn't. Apparently your understanding
of RAA and mine are different. I stated my understanding:

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.


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

Therefore there is *no* gap. Surely you are not going
to tell me that you are okay with the idea of introducing concepts
that lead to contradictions, are you? You know that if we
do that it means our system is then completely unsound,
and we can prove literally anything from that point.

Who knows, between a and a+(a-b)/2 there could be a gap.

If there is, then the system is inconsistent.

a and b are not specific values. If there is a gap
between any two values, then there is a contradiction.
Then: floods, earthquakes, dogs and cats living together
(metaphorically speaking.)

Also: proofs for everyone! Does A imply B? Yes it
does, even without knowing what A and B say. Does
A imply not-B? Yes, it implies that too!

Are you sure it is valid even after finding some
infinite number of "mid-points"?

Yes, I am sure;

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.

- venkat
.

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