Infinite Representation

Personally, I believe in a constructivist approach to mathematics. As such, I do not believe in the notion of completed (actual) infinity: I am a classical finitist, which means I use infinity, but only when it is used in conjunction with a limit. One way in which the use of actual infinity seems problematic to me is related to decimal representation.  In particular, there is a paradoxical statement that is held to be true by infinitists (mathematicians who believe in completed infinity) that numbers can have multiple decimal representations. In their defense, they make the following claim:

Eq 1: 

Most people are not intuitively comfortable with this statement, because it implies that two different numerical (decimal) representations correspond to a single number (and I think that intuition is correct).  The “proof” of identity begins by assigning an infinitely repeating value to a variable x as follows:

Eq 2:

Multiply both sides by 10 to get: 

Eq 3:

Subtract (2) from (3) to get: 

Eq 4:

Equating x in (2) and (4) leads directly to (1).  

Infinitists take the position that this is not paradoxical; they simply accept that a single number has multiple decimal representations, regardless of whether that result is intuitively paradoxical.  

Classical finitists can avoid the paradox by handling infinite representations as limit series.  As a result, believers in potential infinity are not committed to a single number having multiple equivalent decimal representations.

The essential difference in approach is that believers in potential infinity refuse to represent a number which has actual infinite length, so a representation where the nines run “to infinity” is not acceptable.  The potentially infinite representation is therefore written as a limit series:

Eq 5:

In other words, repeating decimal representations are written as a limit where the number of terms approaches infinity.  The significant difference between the limit representation and the repeating decimal representation is that the former is clearly a process, not a number whose length has a cardinality of aleph-null.  

Using the limit representation in (5), equation (4) may be re-written as:

Eq 6:

Note that (6) does not reproduce the result of (4) (i.e., that x=1). 

The different results obtained by using the limit formula as opposed to the overbar formula are the result of being forced to take account of the number of terms in the series expansion.  In other words, we did not take account of the term in the subtraction in (4), because it was assumed that the decimal string was of infinite length.  Therefore, the term i in the second series expansion is subtracted from term i+1 in the first series expansion.


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