Re: well-ordered sets and inductive sets

In article <Pine.BSI.4.58.0510201947450.15396@xxxxxxxxxxxxxxxxx>,
William Elliot <marsh@xxxxxxxxxxxxxxxxxx> wrote:
>On Thu, 20 Oct 2005, ken.quirici@xxxxxxxxxx wrote:
>
>> Are well-ordered sets, using the well-ordering, also inductive
>> sets?
>>
>> well-ordered: partial ordering + every pair of elements has a
>> lesser + every subset has least element
>>
>> inductive: partial ordering + every element a has a successor (I
>> assume this means the successor is a number larger than a
>> and the least such number)
>>
>I suppose a successor is a cover; b covers a when a < b and
> for all x, (a <= x <= b ==> x = a or x = b)
>equivalently
> not some x with a < x < b.
>
>No, they are not the same. The positive integers are well-ordered
>and while all the integers are inductively ordered, they are not
>well ordered

And so you prove that inductive does not imply well-ordered. And yet,
the original question asked whether well-ordered implied inductive.

>> It would seem the answer is yes. For each element take the set of
>> all elements larger. Since the set is well-ordered, this set has
>> a least element.

Only if it is non-empty.

>> This must be the successor.
>>
>> Is that a valid proof or sketch of a proof?

Assuming the definition you have given of "inductive set", assuming
that indeed, "successor of x" means an element y such that x<y and for
all z, if x <= z <= y, then z=x or z=y; then "no." A finite set can
trivially be well-ordered, but the largest element does not have a
successor. You can of course obtain infinite examples by taking any
non-limit ordinal. E.g., take N U {infty}, making infty larger than
every natural number. Then this set is well-ordered, but infty has no
successor.

There are other definitions of "inductive", though they do not seem to
apply to your question. For example, the definition I've always known is:

X is inductive if and only if the empty set is an element of X and
for every x in X, x U {x} is in X;

There is also a notion of "inductive set" in descriptive set theory,
according to Wikipedia.


--
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"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes")
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Arturo Magidin
magidin@xxxxxxxxxxxxxxxxx

.