Modified Dedekind Closure approach

The following is a sort of way to start this, but requires something else.

==Starting from here a possible approach to critique

Consider the set of naturals that has i and j in it, that contains the successor of each element in it, except for j it need not contain the successor.  Such a set exists, the natural numbers.

We can then take the intersection of all such sets, ie the smallest set, or set in common for all such containing sets.

So a containing set is one that contains i, j, contains the successor of each element in the set, except possibly the successor of j.

The natural numbers is such a containing set.

Just as in Dedekind, one can show the intersection of all such containing sets, or common part of such containing sets is also a containing set.

Moreover, no proper subset of it is either. Prove that as exercise.  Look at Dedekind.

Given these sets, we can then apply Dedekind type logic to their properties.  This is like the subspace topology way of thinking if one is familiar with topology.

The closure set so obtained plays a role similar to S in Dedekind’s number paragraphs from 1 onwards really.  For example 36 to 41 numbered paragraphs.  These have to be reinterpreted because the j’ element is not in S, where S is the closure defined above.

The Dedekind work is Nature and Meaning of Numbers, published 1888.

The closure set above we can call the chain or successor chain from i to j.  The containing sets, we can call successor chain containing sets.  The containers are still defined first before the successor chain is defined.  The successor chain is defined as the intersection of the containers, i.e. the common part of the containers.

One has to show the common part is not empty.  Since each container contains i and j, then i and j are in the common part of all the containers.  Similarly, i’ is in the common part if j is not i.  If i’ is not j, then i” is in the common part.

A lemma to prove is that if i to k is a chain and k to j is a chain,
then i to k is in the chain from i to j.

==end of passage to critique

Did we assume that there is  a chain from i to j?

Can we just add that as an assumption?  How?

We could consider the above, but if the closure has the property that it contains the predecessor of every element except for that of i, then it is a chain of succession or gapless set.

Basically, if i is really before j, then the above gives us a chain of succession the way we want.  But if j is before i it gives us something else?  Just the points i and j.  And if these are not successive, then we have a gap in the set.

This is a way to define if i is before j?

Can we just say that there is a set of natural numbers so that i and j are in the set, j is the successor of a number in the set, if it is not i, and possibly some other condition?

Such as each element has a predecessor in the set, except i?

If we impose enough conditions, we will either get a set that exists or no set satisfies the conditions because j is really before i.

We have two strategies.  One is to set up conditions so if i is before j, we get what we want and otherwise get an empty set or some set with a gap.  In this approach we show that if we switch i and j we get the chain we want one way and the empty set or defective set the other way.

A second strategy is we define the set in such a way that if the closure contains the successor of i say or predecessor of j, then i is before j and otherwise j before i.  So we end up that the closure contains i and j, but the successor of one and predecessor of the other, but not vice versa.  This then defines which comes before the other.

Is it easier to just stick to initial segments or final segments first and then do these chains later as an intersection of initial or final segments?

Still, a direct chain way teaches us something.

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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