Dedekind tail sets convenient

Dedekind wants to consider a transform function f(x), or x’ so that it is one to one from the domain to its range.  This will end up being the successor function on natural numbers.

The closed sets under successor are tails of the naturals.  So start from 2 and go on forever is the 2 tail.  When we take the successor of the 2 tail we get the 3 tail.  The 3 tail is also closed. So we have a 1 to 1 function from a closed set to a closed set.

This makes it very easy to work through the small lemmas in Dedekind’s book one after another.  Here we refer to his 1888 book on the natural numbers, “What are numbers and what should they be?”

When we try to use initial segments or head segments from 0 to n or use chains from i to j, we have to deal with the extra complexity. If we use the predecessor function, and we use head segments from 0 to n, then ‘0 will not be defined or we have to start giving it a definition. Either it is 0, which destroys 1 to 1, or we have to start making up extended structures beyond zero. We could call those (-,i). These let us keep the predecessor as 1 to 1. But we now have these extra objects to deal with.

The path Dedekind ended up choosing or constructing avoids these complications.

Advertisements

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.
This entry was posted in Uncategorized. Bookmark the permalink.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s