On 1/19/18 11:00 AM, Jos Bergervoet wrote:
On 1/19/2018 7:19 AM, Richard D. Saam wrote:
On 1/10/18 3:15 PM, Jonathan Thornburg [remove -animal to reply] wrote:
..
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All of the above logic must consider Goedel's incompleteness theorem

First Incompleteness Theorem from Wiki:
"Any consistent formal system F
within which a certain amount of elementary arithmetic
can be carried out is incomplete;
i.e., there are statements of the language of F
which can neither be proved nor disproved in F." (Raatikainen 2015)

or perhaps another statement:

all hypotheses cannot be considered true or untrue
within any axiomatic structure defining the hypotheses.

So, it is impossible to prove or disprove
the complete universe structure
based on any axiomatic structure

Why do you say so? Goedel's incompleteness theorems only
state that *some* statements are not provable. Statements
you refer to here about the universe may not be of that
type!

As a matter of fact, Goedel's own incompleteness theorems
*are* provable. (Using a formal system F of as mentioned,
containing *some* statements are not provable. But
incompleteness theorems still are!)

Taking the words from
Goedel's Proof by Ernest Nagel and James R. Newman
on their analysis of Goedel's 1931 paper:

"If the Principia Mathematica is consistent,
its consistency cannot be established
by any meta-mathematical reasoning that can be mirrored
within the Principia Mathematica itself.
Meta-mathematical arguments establishing consistency
of formal systems such as Principia Mathematica have been devised,
but these proofs cannot be mirrored inside the systems
that they concern."

Goedel apparently tended towards a Platonic view
by which objects had their own identity
i.g. a triangle has its own identity
independent of any logical consistency for it.

Goedel had discussions in this regard with Einstein in the early 1950s
probably in the context of the astrophysical universe.
Surely Einstein had a much more deterministic view.

Other discussions relate to the brain (or life in general)
existing outside of logical consistency.
Goedel would be in the affirmative.

Richard D Saam

[[Mod. note -- I think this discussion has now moved well outside the
subject of this newsgroup ("research in astronomy/astrophysics"). Any
further discussion should probably be over in the sci.math set of
newsgroups. -- jt]]