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Kanovei & Shelah (2004) explicitly constructed a nonstandard extension of the reals. Now, I'm no expert on this but I gather that they first specified a set of free ultrafilters, and then used it to explicitly specify a particular free ultrafilter, which they used to construct a nonstandard extension of the reals. Here's the theorem they proved:

In ZFC, there exists a definable, countably saturated extension $^*\mathbb{R}$ of the reals $\mathbb{R}$, elementary in the sense of the language containing a symbol for every finitary relation on $\mathbb{R}$.

They said about their construction:

A somewhat different, but related problem of unique existence of a nonstandard real line $^*\mathbb{R}$ has been widely discussed by specialists in nonstandard analysis. Keisler notes...that for any cardinal $\kappa$, either inaccessible or satisfying $2^\kappa=\kappa^+$, there exists a unique, up to isomorphism, $\kappa$-saturated nonstandard real line $^*\mathbb{R}$ of cardinality $\kappa$, which means that a reasonable level of uniqueness modulo isomorphism can be achieved, say, under GCH. [Our theorem] provides a countably saturated nonstandard real line $^*\mathbb{R}$, unique in absolute sense by virtue of a concrete definable construction in ZFC.

The model they constructed is unique in the sense that given their definitions, one and only one model can be constructed. Here's my question: given a different set of definitions, can a totally different nonstandard extension of $\mathbb{R}$ be explicitly constructed via the same strategy they used? I'm just wondering whether more nonstandard models can be explicitly defined, other than the one defined by Kanovei & Shelah. Perhaps if a different set of free ultrafilters is specified from the start, then a different nonstandard extension can be constructed.

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  • $\begingroup$ I don't understand how "given a different set of definitions" and "via the same strategy they used" fit together. What do you see as the strategy vs. the definitions used? $\endgroup$ Nov 17, 2021 at 23:54
  • $\begingroup$ Here's what I think their strategy is: Specify a set of free ultrafilters, then use that set of free ultrafilters to specify a particular free ultrafilter, which is used to construct a nonstandard extension of the reals. By "definitions", I mean the set of free ultrafilters specified in the first step. If the first step is different, then perhaps the extension will be different too. But the overall strategy is still the same. Does this help? $\endgroup$
    – phst
    Nov 18, 2021 at 0:04
  • $\begingroup$ To further clarify, when I said "the first step is different", I meant the set of free ultrafilters specified is different from the one specified by Kanovei & Shelah. The general steps of their construction will remain untouched but the details will change. This is what I mean. $\endgroup$
    – phst
    Nov 18, 2021 at 0:17

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