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Infinity Does Not Exist
9 Jan 2006
By Torgny Tholerus
Introduction
All mathematicians and all philosophers believe that infinity exists. But they are all wrong. I shall explain why in this paper.
Universal quantifier
What does the universal quantifier mean? What does it mean when we say (x)P(x) - (For all x, x have the property P)?
One reasonable interpretation is that for all elements in the set of all objects in universe, that element has the property P. We then define the set of all objects in universe as containing all past, present and future objects in the universe, including all abstract objects such as all the natural numbers, all the real numbers, all sets, and all subsets, etc.
This set of all objects in universe is a fixed set it can not be changed or extended. If (x)P(x) is true yesterday, it must also be true tomorrow.
So the meaning of (x)P(x) is such that you can deduce from that that P(A) is true if A belongs to the set of all objects in universe.
The Russell set
Now let us construct the Russell set R consisting of all sets that are not members of themselves. We start from the set of all objects in universe and from those we select the objects that are sets and which are not members of themselves.
Is R an object in universe? Is R a member of the set of all objects in universe?
Then suppose first that R does not contain itself. From that we deduce that R contains itself, which is a contradiction. So R must then contain itself. But from that we deduce that R does not contain itself. So we will then also get a contradiction.
From this we can conclude that R is not an object in universe. R is not a member of the set of all objects in universe. And because the set of all objects in universe is a fixed set, R can not be added to this set. So R must be an object outside universe.
A further conclusion is that R is not a member of itself. This is not a contradiction because R was not included in the set from which the elements of R were selected.
Now it is proved that there exist both objects inside universe and objects outside universe.
Outside Universe
But how do we decide if an object A is inside or outside universe? What is the demarcation line between objects inside universe and objects outside universe?
One possible rule for this is to look at how the object A was construed. If the only way to construct A is to start from all objects in universe, then A will most likely be a new object, an object that did not exist before, A will not be any of the objects inside universe.
So there is some sort of temporal difference between the objects. The objects inside universe are the objects that existed before. And the objects outside universe are new objects that exist after.
But this means that the predicate logic conclusion must be modified. The new formulation is:
(x)P(x) % P(A), if A is an object inside universe.
If A is an object outside universe, then the conclusion P(A) is invalid and illegal.
One further conclusion is that the set of all objects in universe is itself an object outside universe, because this set is construed from all objects in universe. So this set will not be a member of itself. The same is true for the set of all sets, because this set is construed from all objects in universe by selecting those objects that are sets. This set is an object outside universe and it is not member of itself.
This is a general rule. No set can be a member itself, because the set does not exist before it has been construed. So the set R of all sets that are not members of themselves is exactly the same as the set of all sets.
Natural numbers
Now look at the set of all natural numbers. This set is construed from all objects in universe, by selecting all objects that are natural numbers. So the set of all natural numbers is a new object, it is an object outside universe.
Now suppose that there exist a biggest natural number M in this set. What will happen if we construct the successor of M, M+1?
In this case the number M+1 is a natural number. But M+1 does not belong to the set of all natural numbers, because M+1 is an object outside universe. This is OK. So there is no contradiction. M is still the biggest number in the set of all natural numbers.
So there exists a biggest natural number M. M is an object inside universe, and M+1 is an object outside universe.
But what will happen if we define a set N such that: For every number n in N there will be a successor n+1 that also is a member of N?
But this definition will be illegal because it is a circular definition, when we define N we are presupposing that we have N to start with. This is not allowed. So we have to reformulate this definition as: For every natural number n, if n is a member of N then also the successor n+1 will also be a member of N. This will be a legal definition because in this case we start from all objects in universe and from that construe the set N.
So in this case the definition says that: For all natural number n from 1 to the biggest number M, if n is a member of N then n+1 will be a member of N. So N will be the set consisting of all natural numbers plus the number M+1. And this set N will then also have a biggest number M+1.
So the set of all natural numbers is finite. There is no mapping from all natural numbers to a strict subset. The biggest natural number M will mostly have no mapping.
So more precisely, what is the set of all natural numbers? One possible definition is all explicit natural numbers given by any human in the past and in the future. Amongst all these numbers there will be a biggest one, but we cannot say exactly what number. The only thing we can say is that there exists one.
Conclusion
In order to be able to use the universal quantifier we have to define what is meant by it. Then we need the set of all objects in universe. This set is a fixed set that can not be changed.
Then we looked at the Russell set R, and proved that this set was an object outside universe. Then we identified the objects outside universe as objects created out of the set of all objects.!#$%-ABCOP^
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Torgny Tholerus
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