FOM: Steve Simpson on K"onig's lemma, non-standard models etc.

[Neil Tennant]

Steve Simpson writes (in response to my question whether the existence
of non-standard models for full arithmetic is strictly weaker than the
completeness and compactness theorems):

_____________________
The existence of a nonstandard model M of arithmetic
implies the existence of an omega-model of weak K"onig's lemma, namely
the Scott system of M.  And the G"odel completeness and compactness
theorems hold in any such omega-model; in fact, they are equivalent to
weak K"onig's lemma, in the sense of reverse mathematics.  All of this
is provable in weak systems, e.g. RCA_0.  
_____________________

Later he adds:

_____________________
However, this is not quite an airtight answer to Neil's question,
because the existence of a nonstandard model of arithmetic does not
literally imply weak K"onig's lemma.  It only implies the existence of
an omega-model of weak K"onig's lemma, as noted above.  Nevertheless,
I would be willing to (somewhat loosely) summarize the precise results
above as follows: The existence of nonstandard models of arithmetic is
equivalent to the G"odel completeness and compactness theorems.
_____________________

The not-completely-airtight answer has I think been too loosely
summarized. Trusting Steve's precise technical results, isn't their
correct summary only that

	The existence of nonstandard models of arithmetic implies
	the consistency of assuming that completeness and compactness hold.

...?

Maybe I'm missing something here by not having all the succinct
technical phrases fully defined. Steve, could you clarify the position
and/or correct me?

Neil Tennant