August 7th, 2018, 03:48 PM  #11  
Senior Member Joined: May 2016 From: USA Posts: 1,148 Thanks: 479  Quote:
 
August 8th, 2018, 12:53 AM  #12 
Senior Member Joined: Jun 2015 From: England Posts: 884 Thanks: 265  
August 8th, 2018, 06:00 AM  #13 
Senior Member Joined: May 2016 From: USA Posts: 1,148 Thanks: 479  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, 06:15 AM  #14  
Math Team Joined: May 2013 From: The Astral plane Posts: 1,888 Thanks: 767 Math Focus: Wibbly wobbly timeywimey stuff.  Quote:
Dan  
August 8th, 2018, 07:06 AM  #15  
Senior Member Joined: Jun 2015 From: England Posts: 884 Thanks: 265  Quote:
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, 07:17 AM  #16  
Senior Member Joined: May 2016 From: USA Posts: 1,148 Thanks: 479  Quote:
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 07:20 AM.  
August 8th, 2018, 08:09 AM  #17 
Senior Member Joined: Nov 2011 Posts: 230 Thanks: 2  
August 8th, 2018, 08:35 AM  #18  
Senior Member Joined: May 2016 From: USA Posts: 1,148 Thanks: 479  Quote:
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This not captious. The OP was unclear in what was being asked and unclear what answers would be acceptable.  
August 8th, 2018, 08:44 AM  #19  
Senior Member Joined: May 2016 From: USA Posts: 1,148 Thanks: 479  Quote:
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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