Amorphous set with injection distinct tuples evens to odds
David Libert
[1] David Libert Mar 16 '03 "Re: "Bi-surjectiveness""
http://mathforum.org/discuss/sci.math/m/490333/491228
[2] David Libert Mar 17 '03 "Re: "Bi-surjectiveness""
http://mathforum.org/discuss/sci.math/m/490333/491250
In the base of that thread
[3] Mar 14-17 '03 13 articles "Re: "Bi-surjectiveness""
http://mathforum.org/discuss/sci.math/m/490333
the original poster Rolf Bardeli asked whether two sets A and B
having surjections must be isomorphic, and related questions. Various
articles in [3] noted ZFC proves bi-surjective sets are isomorphic.
In the parent article of [1], Herman Rubin noted that for U_1
amorphous the sets A the set of positive even length tuples of
distinct U_1 elements and B the set of odd length tuples of distinct
U_1 elements are bi-surjective but non-isomorphic.
The two surjections are obtained by projection by deleting last
coordinate.
[1] followed up Herman's article, noting in more detail why A and B
are non-isomorphic. [2] added a technical correction to [1].
[1]-[2] proved that for such A,B, at least one direction of
injectibility between them must fail. [1]-[2] also gave two Fraenkel
Mostoski ZF models with alternate example of such A,B, (ie the
respective A,B defined over alternate examples of U_1 's), one
example where both directions of injectibility failed, and a second
example where B could inject into A (ie odds to evens). Hence, by
the previous result, in this second example A can't inject into B.
[1] raised but did not answer the question of whether there could be
another example with A injecting into B (ie evens into odds).
The new result of this present article is a Fraenkel Mostowski ZF
model with an amorphous U_1 with A injecting into B (ie evens to
odds), answering affirmatively that question from [1].
The construction method below, to do this, is similar to the
construction of an amorphous set with a group structure from
[4] David Libert Feb 3 '03 "Re: Group Structure on any set"
http://mathforum.org/discuss/sci.math/a/m/478403/479486
Let V be a vector space with countable basis over the field of
scalars Z/2. I will make a Fraenkel Mostowski ZF model, with set of
atoms corresponding to the members of V. The permutation group on the
atoms will be the automorphism group of V as a Z/2 vector space, with
the usual action of that group. The FM model will be finite support.
We will be considering V+, the non-0 elements of V, as a set
inside the FM model.
Given any finite subset F of V+, we can consider the subspace of V
this generates, also a finite set since Z/2 is finite. Given any pair
of vectors outside of that subspace, we can find an automorphism of V
fixing the subspace and sending one such vector to the other. This is
enough to get V+ is amorphous in the FM model. (A similar argument was
mentioned in [4]).
The vector space structure of V is respected by the automorphism
group, hence it has empty support in the FM construction, so it makes
it into the FM model.
I will use that vector space structure to define in the FM model an
injection : A >-> B ie positive even length tuples of distincts into
odd length tuples of distincts.
So suppose <v1, v2, ... , vn> is a tuple of distinct V+ members,
with n positive even. I seek to define the associated odd length
tuple of distinct V+ members.
Consider the subspace generated by {v1, ... vn}. We define a basis
for this subspace from the <v1, ... , vn> tuple, using the tuple
ordering between coordinates to do so. (Ie tuples in permuted order
of each other can get different bases this way). Namely work through
v1, v2 , ... vn in turn, throwing into the growing basis any element
not in the span of the part of this basis previously added.
We obtain a basis of m elements, for some m : 1 <= m <= n . Using
that basis, we can associate each non-0 vector in the the subspace
generated by {v1, ... vn} an m-tuple of Z/2 elements, ie the
coordinates of the vector realtive to this basis.
We can linearly order the set of such m-tuples, lexicographically on
a linear ordering on Z/2.
{v1, ... , vn} is a subset of that subspace, indeed of the non-0
part of that subspace. The subspace has size 2^m (m basis vectors
over Z/2), an even number. So removing the 0 vector, the non-0 part
of the subspace is odd size. That odd sized set is a superset of the
starting even sized set {v1, ... , vn}, so the inclusion is proper.
So we can find the least element of the non-0 part of the subspace,
according to the linear ordering we defined, which is not a member of
{v1, ... , vn}, such must exist by the properness result.
We have found a way to definably pick a member of V+ outside of
{v1, ... , vn}, definable using only the vector space structure on V,
which is available in the FM model.
Letting vn+1 be the V+ member so selected, we map <v1, ... , vn>
to <v1, ... , vn, vn+1>, ie append vn+1, so mapping the even
length tuple <v1, ... vn> to an odd length tuple.
The inverse of this map is just by deleting last coordinate, so this
map is injective.
So V+ is the desired amorphous U_1 with A >-> B.
--
David Libert ah...@FreeNet.Carleton.CA