My Math Forum zero in the middle of x-axis

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August 7th, 2018, 04:48 PM   #11
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Quote:
 Originally Posted by Maschke You can't and it's not. Consider a line, infinitely long in each direction. Like Euclid's idea of a line. Now pick an entirely arbitrary point on the line and call it 0. Pick another point on the line and call it 1. Now having chosen those two arbitrary points, you can define all the integers, positive and negative whole numbers, based on the unit length between 0 and 1. Then you can define the rationals, all the fractions between the integers. And then you can define the rest of the real numbers by plugging the holes where there are points but not rationals. So you see that the labelling of the points is completely arbitrary. 0 and 1 could be anything you like. Here's another way to look at it. You have a bunch of houses on the street. One day the city council decides to renumber all the houses. The addresses of the houses all change. But the houses themselves remain exactly the same as they were before. A real number is the address of a point on a line. That's all it is. A real number is an address, not the point itself. And addresses are arbitrary, they don't tell you anything about the points themselves. Likewise in a computer program. The address of a variable is not the same as the contents. The address is just the memory location where the value of some variable lives. The address is not the variable. It's only the address.
And yet there are an equal number of points to the left of the arbitrarily chosen zero as to the right.

August 8th, 2018, 01:53 AM   #12
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Quote:
 Originally Posted by JeffM1 Somehow we seem to be getting back to definitions.
Can't say I understand your comment?

August 8th, 2018, 07:00 AM   #13
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 Originally Posted by studiot Can't say I understand your comment?
Studiot

The very first response says that a proof is possible only if we have definitions and axioms.

So why does the "finite" necessarily lie "between" something called "infinity" and something called "negative infinity? What do those words mean?

Perhaps our definitions and axioms allow us to say:

$x \text { is finite } \iff x \text { is not transfinite.}$

$x \text { is between } a \text { and } b \iff a < x < b \text { or } b < x < a.$

$\infty \text { is transfinite.}$

$-\ \infty \text { is transfinite.}$

$-\ \infty < \infty.$

$x \text { is finite} \implies -\ \infty < x < \infty.$

Using standard rules of inference, we can now prove the following theorem:

$x \text { is finite } \implies x \text { is between } -\ \infty \text { and } \infty.$

Now, given all that, the requested proof is trivial once we have in addition the proposition that "zero is not transfinite." But is that proposition an axiom or a theorem? If the latter, where is its proof?

The OP asks for a proof without specifying definitions, axioms, or rules of inference. In that vague context, "zero is finite" is merely an unproved assertion about undefined terms and does nothing to clarify what "between" means.

August 8th, 2018, 07:15 AM   #14
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Quote:
 Originally Posted by JeffM1 Studiot The very first response says that a proof is possible only if we have definitions and axioms. So why does the "finite" necessarily lie "between" something called "infinity" and something called "negative infinity? What do those words mean? Perhaps our definitions and axioms allow us to say: $x \text { is finite } \iff x \text { is not transfinite.}$ $x \text { is between } a \text { and } b \iff a < x < b \text { or } b < x < a.$ $\infty \text { is transfinite.}$ $-\ \infty \text { is transfinite.}$ $-\ \infty < \infty.$ $x \text { is finite} \implies -\ \infty < x < \infty.$ Using standard rules of inference, we can now prove the following theorem: $x \text { is finite } \implies x \text { is between } -\ \infty \text { and } \infty.$ Now, given all that, the requested proof is trivial once we have in addition the proposition that "zero is not transfinite." But is that proposition an axiom or a theorem? If the latter, where is its proof? The OP asks for a proof without specifying definitions, axioms, or rules of inference. In that vague context, "zero is finite" is merely an unproved assertion about undefined terms and does nothing to clarify what "between" means.
I like the conversation and all but you do realize this is High School Algebra?

-Dan

August 8th, 2018, 08:06 AM   #15
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Quote:
 Originally Posted by JeffM1 Studiot The very first response says that a proof is possible only if we have definitions and axioms. So why does the "finite" necessarily lie "between" something called "infinity" and something called "negative infinity? What do those words mean? Perhaps our definitions and axioms allow us to say: $x \text { is finite } \iff x \text { is not transfinite.}$ $x \text { is between } a \text { and } b \iff a < x < b \text { or } b < x < a.$ $\infty \text { is transfinite.}$ $-\ \infty \text { is transfinite.}$ $-\ \infty < \infty.$ $x \text { is finite} \implies -\ \infty < x < \infty.$ Using standard rules of inference, we can now prove the following theorem: $x \text { is finite } \implies x \text { is between } -\ \infty \text { and } \infty.$ Now, given all that, the requested proof is trivial once we have in addition the proposition that "zero is not transfinite." But is that proposition an axiom or a theorem? If the latter, where is its proof? The OP asks for a proof without specifying definitions, axioms, or rules of inference. In that vague context, "zero is finite" is merely an unproved assertion about undefined terms and does nothing to clarify what "between" means.
Well yes if you introduce more new terms than the OP did, you have more of a proof problem.

Neither I nor the OP mentioned transfinite.

Nor I did offer a complete proof.

The OP said "How do I prove......"

Which implies that he wanted a pointer in the right direction and I tried to offer this, in terms of what was already said and presumably set as a course question.

So the question mentioned infinity

So is zero infinite?

Taking the Dedekind definition two things are infinite - positive and negative infinity.

Is zero either of these?

No
So by, Dedekind's definition, anything else is defined as finite so (the number) zero is finite.

The question asks is zero between?

Well there are three possibilities in those terms.

1)zero is beyond plus or minus infinity.
But there is nothing beyond plus or minus infinity that is null or void.
So zero is not beyond plus or minus infinity.

2) zero is at plus or minus infinity.
We have just proved it is not

That leaves
3) zero is between plus and minus infinity

August 8th, 2018, 08:17 AM   #16
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 Originally Posted by topsquark I like the conversation and all but you do realize this is High School Algebra? -Dan
I tutor high school algebra, and proofs are notable for their absence. I admit that perhaps I am being uncharitable, but even at the high school level, this question is either very trivial or very badly posed.

Let's assume that we are talking about integers.

Ideas that are likely to be assumed by a high school student.

$0 \in \mathbb Z.$

$x \in \mathbb Z \implies -\ \infty < x < \infty.$

$x \text { is between } a \text { and } b \implies a < x < b \text { or } b < x < a.$

Now if this is the set of ideas being worked with, the proof of whether zero is between negative infinity and infinity is trivial.

But perhaps this is a very interested and clever student who is asking whether the number of integers greater than zero equals the number of integers less than zero. The answer (if there is a meaningful answer) is intuitive, but the proof involves introducing some concepts that are not taught in most high school algebra classes and that were not even formalized before the late nineteenth century. If we are finitists, the question is meaningless. Otherwise, we need to specify what we mean by the equality of the number of two sets.

Or perhaps the OP is asking for a proof that zero is the only integer such that the number of integers less than itself equals the number of integers greater than itself. Now perhaps we can construct a consistent mathematics where that proposition is true, but it is false in terms of Cantor's definitions.

Until the OP clarifies what in the world is being asked and how rigorous a proof is required, any answer will do.

I have clearly been wrong in being too flippant in my answers and so have created unnecessary misunderstandings, but the very first response to this post was right on the money: what exactly does the OP mean by the words in the question posed and what assumptions is the OP willing to accept in a proof. I do not think that is an unfair demand. Indeed, for those students who are interested in proofs, understanding that careful definitions are the first step in a proof is salutary.

Last edited by JeffM1; August 8th, 2018 at 08:20 AM.

August 8th, 2018, 09:09 AM   #17
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Quote:
 Originally Posted by JeffM1 I t ...right on the money: what exactly does the OP mean by the words in the question posed and what assumptions is the OP willing to accept in a proof.
Z+ = Z-
because of Z+ = N = Z-

August 8th, 2018, 09:35 AM   #18
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Quote:
 Originally Posted by studiot ... Neither I nor the OP mentioned transfinite.... So the question mentioned infinity
So is infinity transfinite?

Quote:
 Taking the Dedekind definition
But you did not understand why I said that it comes back to definitions. In that case, why are you bringing in definitions?

Quote:
 But there is nothing beyond plus or minus infinity that is null or void.
Hmm, are we now back to assertions that Cantor's diagonal proof is invalid?

This not captious. The OP was unclear in what was being asked and unclear what answers would be acceptable.

August 8th, 2018, 09:44 AM   #19
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Quote:
I do not agree that the set of natural numbers equals the set of negative integers.

If you define how to determine the equality of the numbers of two sets what the number of a set is, then you can prove that the number of the set of natural numbers equals the number of the set of negative integers. But you have not proved that statement: you have merely asserted it.

Moreover that is not a property unique to zero if that was what you were originally trying to prove. It is true of every integer: that the number of integers greater than a given integer equals the number of integers less than the given integer is a proposition true for any integer.

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