Ordinal Fractions von Neumann style

We can take the von Neumann construction of whole numbers in logic and set theory and apply the same style to fractions.  This is an ordinal concept, in particular where order is determined by set membership.

{0}

1 ={0}

2 = {0,1} =  {0,{0,1}}

..

We can think of this as applying along each bin line.

Along a bin line we have

0,1,2,3

With two bin lines in parallel, such as 1 and 2, we have

0,1;2,3;4,5

0,1,2,

We can add the bin label as follows

(0,2),(1,2);(2,2),(3,2);(4,2),(5,2)

(0,1),(1,1),(2,1),

We can then insert, if we like, the von Neumann lists in for one or both numbers in the pairs.  We can think of the bin number or bin index as a label.

To compare two fractions ordinally, we find a common bin line, ie common denominator bin line and we then use ordinality along that line.

So to compare 1/3 and 1/2, we convert them to 2/6 and 3/6, which are the numbers 2 and 3 along the bin 6 line.  We then do the standard ordinal comparison.

We don’t need to use a von Neumann construction for the ordinal numbers.  We can use any construction for them, or just think of the above as an ordinal construction or ordinal tableau.

We find a common line to do ordinal comparisons.  The multiple lines together form a sort of ordinal system of numbers which we can call ordinal fractions.  The system of lines lets us compare fractions by ordinal methods.

Set membership and subset is how the von Neumann construction constructively does ordinal comparisons.  Once we get to a common bin line, say bin 6 for comparing 1/2 and 1/3, we can then apply the von Neumann construction or whatever ordinal construction we use.

We can also think of combining bin lines, using the bin marker.  So

(0,1),(1,2),(1,1)

Is 0, 1/2 and 1

(0,1)(1,3)(1,2)(2,3)(1,1)

Is 0, 1/3, 1/2, 2/3 and 1.

We can put these into lists as sets if we want.

1/3 = {(0,1)}

1/2 = {(0,1),(1,3)}

2/3 = {(0,1),(1,3),(1,2)}

So 1/3 is less than 1/2, because of set membership.

We can think of different types of constructions to produce ordinal relationships by set membership.  The ones given here may not be the best ones or best way to express it.  But this gives an idea of how we can generalize von Neumann type lists to fractions and then do ordinality based on set membership.

The different bin lines and their labels then give us the means to mix and match in various ways in order to get the effect of ordinality by set membership.

 

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About New Math Done Right

Author of Pre-Algebra New Math Done Right Peano Axioms. A below college level self study book on the Peano Axioms and proofs of the associative and commutative laws of addition. President of Mathematical Finance Company. Provides economic scenario generators to financial institutions.
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