Talk:Whitehead problem

Logic question

I have what is probably a stupid question:

• if the statement is false, then you could describe a group A that refutes the statement.
• the statement is undecidable implies that the statement cannot be refuted which implies that there is no such group A
• doesn't this imply that the statement is true?

Am I failing in my initial assumption that you are able to describe any group? Or perhaps in assuming that you can decide based on the description of a group whether it is free? --Radu Raduberinde (talk) 2008-07-16T21:01:54

This is more a question of logic, than of abelian groups. I think perhaps it is best understood as not being able to decide if a particular description describes a free group. Another view is that we do not care particularly whether or not our statements are true or false, only if they can be deduced from our assumptions. The assumptions of ZFC are neither sufficient to deduce that all whitehead groups are free, nor that there is a whitehead group that is not free.

Neutral geometry is similar. It does not include enough assumptions to decide on one particular matter, so it is able to model a variety of more specific situations, including those where the particular matter is settled as "true" and those where the particular matter are settled as "false". JackSchmidt (talk) 21:11, 16 July 2008 (UTC)

What "undecidable in ZFC" means is that there is one model of ZFC in which every Whitehead group is free, and another model of ZFC in which there is a Whitehead group that is not free. Keep in mind that whether a group is a Whitehead group from the point of view of a model of set theory depends not only on the group's internal structure but also on the collection of other Abelian groups in the model and the collection of group homomorphisms in the model. — Carl (CBM · talk) 01:29, 17 July 2008 (UTC)

Thanks! --Radu —Preceding unsigned comment added by Raduberinde (talk • contribs) 14:57, 17 July 2008 (UTC)

Different uses of "Whitehead group"

Maybe one should add the fact that every free abelian group is Whitehead?

Also, there are different uses of the term "Whitehead group" which might lead to confusion. Some authors, among them the Wikipedia list of ZFC undecidable statements, call "Whitehead groups" only the non-free abelian groups with Ext(W, Z)=0. I think the article should mention this ambiguity.

If nobody disagrees I will add these points. Ninte (talk) 07:25, 29 June 2009 (UTC)