[FOM] Elements of Librationism and the Continuum Hypothesis

[Frode Bjørdal]

My paper *Elements of Librationism* is now Arxived (
http://arxiv.org/abs/1407.3877) and has been submitted. A main result is
that librationism (£) interprets ZFC if what we call the Skolem-Fraenkel
Postulate holds; SFP is near the assumption that replacement with first
order functional conditions does not create paradoxes given the other
assumptions of ZF, but our operation *librationist capture* is slightly
stronger and entails collection and choice in the desired contexts.

This posting can in part be understood as a follow up to earlier postings
of mine meeting a challenge by Harvey Friedman to post alternative
foundational approaches on this email list.

The abstract at arxiv is as follows: We develop librationism, £, and
clarify some mathematical and philosophical matters which relate to the
particular manner in which it deals with the paradoxes and to its
mathematical usefulness. We isolate a domination operation which unlike the
power set operation is not paradoxical and which helps us isolate the
definable real numbers. Under a plausible postulate we show that £
interprets ZFC; our strategy for achieving this involves extending an
interpretation of ZF in a system weaker than ZF with collection minus
extensionality by Harvey Friedman, and a novel notion of librationist
capture which entails collection, specification and choice in desired
contexts.

In future updates I will point out that we can force generic extensions in
a very facile manner in the set up we have for what we call the definable
echelon or in related contexts, and we may also to an unknown extent expand
with inaccessible numbers. In the presented definable echelon the continuum
hypothesis will hold according to £ as the structure is minimal, but one
may clearly force width-extensions where it does not hold. To my mind this
does not suggest that the continuum hypothesis is vague, but rather that it
is not precise. Notice well that all talk about uncountable sets in £ is
only relative to the functions available in the structure under
consideration, and that £ outside that structure has a function from the
natural numbers onto any set in that structure under consideration; on
this, cfr. Skolem's "paradox".


..........................................
Professor Dr. Frode Bjørdal
Universitetet i Oslo Universidade Federal do Rio Grande do Norte
quicumque vult hinc potest accedere ad paginam virtualem meam
<http://www.hf.uio.no/ifikk/personer/vit/fbjordal/index.html>
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