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Expansion Form and an ImplicationPhilip White

There is a common model for the naturals referred to here as the natural expansion, andsimilarly, there is a common model for the reals referred to here to as the real expansion. Assuming either of these denumerably infinite models, it will be shown that a certain ambiguity arises whenevaluating what these models represent.

Def. 1: Base-10 Natural Expansion Form Consider all of the numbers represented by the following model …x 4x3x2x1; where the number of x i placeholders is countably infinite, and where there is a set S={0,1,2,3,4,5,6,7,8,9} such that each x i must by any s S. Numbers captured by this model are said to be in base-10 natural expansionform or alternatively B10 .

Def. 2: Base-10 Real Expansion Form Consider all of the numbers represented by the following model …x 4x3x2x1.y 1y 2y 3y 4…; where thenumber of x i , y i placeholders is countably infinite, and where there is a set S={0,1,2,3,4,5,6,7,8,9}such that each x i , y i must by any s S. Numbers captured by this model are said to be in base-10real expansion form or alternatively B10 .

Concerning Def. 1, if a number is written like 351, or 1000, assume this is shorthand for anumber captured by B10 . So when a number is written like 999, or 29, there actually exists animplicit but infinite string of zeros to the left of these finite ideographs. Now, as can be seen, every natural number written fits the model given in Def. 1, and this is done without any double counting.

Concerning Def. 2, if a number is written like 3.45, or 500.000, assume this is shorthand fora number captured by B10 . When a number is written like 3.1, or 999.999, there exists an implicityet infinite string of zeros to both the left and the right of these finite ideographs. It is true thatB10 counts infinitesimals twice (.99999…=1.000…, etc), but this shouldn’t make a difference inthe long run. For now, notice every real number (whether double counted or not) fits the modelgiven in Def. 2.

Lem.Notice both B10 and B10 model some numbers. There exists a bijection between the numbersmodeled by B10 or B10 respectively, and some corresponding infinite ordinal.

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Just which ordinals are bijectable with B10 or B10 is a later topic, but the generalstatement of Lem. is shown below.

Proof.Concerning Def. 1 ( B10 ), notice there are a countably infinite number of x i placeholders.

Similarly, concerning Def. 2 ( B10 ), notice there are a countably infinite number of x i , y i placeholders. Recall the set S={0,1,2,3,4,5,6,7,8,9}. Because there are an infinite number of x i and xi ,y i placeholders respectively, and because each placeholder must be some s S, each model B10

and B10 captures an infinite list of numbers. Now suppose that that α is the first finite ordinalbijectable with the numbers captured by either B10 or B10 . This would imply that at least one of the two models captures only a finite set of numbers, but it was just stated that each model capturesinfinitely many numbers. Thus we have a contradiction, and this implies that the captured numbersare bijectable with some infinite ordinal. □

Now, moving on from the more general proof above, a puzzling situation will be presented. The next task is to calculate the exact cardinality of the numbers counted by B10 or B10 ,however this will be harder than expected. Consider the following conjecture, and afterwards twomethods will be given.

Conj.Going with the definition of B10 and B10 above, it will be shown that infinite natural or infinitereal expansions cannot accurately capture the naturals or the reals.

Method 1.First investigate the numbers counted in Def. 1. Because Def. 1 counts the naturalnumbers, it should follow that the set of numbers captured by B10 should have a cardinality of 0.Say that each |x i |=10. Because 10 is a finite cardinality, and because an infinite product of countable cardinalities equals 0, the cardinality of the numbers implied by Def. 1 is…|x 3|*|x 2|*|x 1| = …10*10*10 = 0 or something like that. So the set of natural numberscaptured by B10 has a cardinality of 0, and this implies that B10 captures a countably infinite listof natural numbers – which seems pretty uncontroversial.

But the same logic used in the preceding paragraph can be applied to Def. 2, but B10

counts the reals instead of the naturals. Remember that each |x i |=10 and each |y i |=10, which

implies that the cardinality of the set of reals captured by B10 can be calculated below:

…|x 3|*|x 2|*|x 1|.|y 1|*|y 2|*|y 3|… = …10*10*10*10*10*10… = 0

But this makes no sense! Early it was stated that Def. 2 captures every imaginable realnumber. How can there be only 0 real numbers? However, if the numbers captured by B10

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cannot be counted with infinite accuracy, then perhaps the same must apply to B10 , and then evenan infinite list of natural numbers becomes questioned – which is unattractive. ■

Method 2. Start with investigating Def. 2. Method 1 states that B10 could be calculated like

…10*10*10*10*10*10…, but then consider the following: If 10=101

, 10*10=102

, 10*10*10=103

,and 10*10*10*10=104 ad infinitum, then perhaps the upper bound of this sequence is 10 0=2 0. Thisimplies that the cardinality of the numbers counted by B10 is 2 0, which makes sense becauseB10 counts the real numbers (for now).

But the same logic used in the preceding paragraph could be applied to Def. 1. This wouldimply that there are a 2 0 number of natural numbers – so similar to Method 1, Method 2 arrives at aproblem. 0 is the accepted cardinality for the natural numbers, how could this cardinality be 2 0?

Therefore, if the cardinality of the numbers captured by B10 cannot be captured to an infiniteaccuracy, then the numbers captured by B10 cannot be discerned either – and therefore an infinite

decimal becomes impossible. ■

The results of this investigation are confusing even to myself, as I am not a seasonedmathematician. Are the arguments presented here a case for finitism? Or perhaps the argumentshere prove that real numbers are not captured by infinite decimals (in the sense of mathematicalPlatonism)? Either way, the above proof introduces an ambiguity when talking about the cardinality of the infinite sets or . Anyhow, there is more research on the way.© 03/08/2013. Philip White.