Set Theory – Meaning and Belief in the Axiom of Replacement

lo.logicmathematical-philosophyset-theory

Here Professor Blass describes the following cumulative hierarchy of sets:

Begin with some non-set entities called atoms ("some" could be "none" if you want a world consisting exclusively of sets), then form all sets of these, then all sets whose elements are atoms or sets of atoms, etc. This "etc." means to build more and more levels of sets, where a set at any level has elements only from earlier levels (and the atoms constitute the lowest level). This iterative construction can be continued transfinitely, through arbitrarily long well-ordered sequences of levels.
This so-called cumulative hierarchy is what I (and most set theorists) mean when we talk about sets.

We want to agree on the following principles:

For every level there is a succeeding level.
For every sequence of levels: $l_1,l_2,l_3,\dots$ there is a level succeeding all levels $l_1,l_2,l_3,\dots$. One might call this level "limit level".

Question:

Why is the axiom of replacement true under this interpretation of the term "set" (set = anything that is formed at some level of this hierarchy)?

Best Answer

For an argument that the iterative conception implies something weaker than unrestricted Separation (implied by unrestricted Replacement), i.e. $\Sigma_2$ Replacement, see Randall Holmes 2001 http://math.boisestate.edu/~holmes/holmes/sigma1slides.ps. (According to Professor Holmes, “this contain[s] an error, which Kanamori pointed out to me and which I know how to fix.”)

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[Math] Intended interpretations of set theories

While Kunen takes for universe the collection of all hereditary sets, Marc proposes to restrict the universe to those hereditary sets which are first-order definable without parameters (Marc's "provable sets"). To make this more mathematical, let me rephrase the question as follows:

Suppose $M$ is a model of ZF. Does the subset $K$ of $M$ consisting of the first-order definable (without parameters) elements of $M$ form a model of ZF?

This turns out to be a delicate question which was answered here on MO some time ago. To summarize the answer there, Marc's proposed universe $K$ will be a model of ZF if and only if $M$ is a model of ZF + V = OD.

Returning to the more philosophical question, this says that Marc's proposal is tenable if and only if there is a reason to believe that the hereditary sets are all ordinal definable. Occam's Razor type arguments support that V = OD and, indeed, V = L is true. However, since it is much weaker than the rigid assumption V = L, the assumption V = OD is not incompatible with a much richer view of the universe...

There is another interpretation of Marc's "provable sets." Instead of merely requiring them to be first-order definable without parameters, one can require them to provably exist in any well-founded model of ZF. With this interpretation, one gets a potentially much smaller set: the minimal well-founded model of ZF. This minimal model is $L_\alpha$ where $\alpha$ is the smallest element of $Ord\cup\{Ord\}$ such that $L_\alpha \models ZF$.

Note that it is entirely plausible that $\alpha = Ord$ (for example this will happen if we happen to live in the minimal model in question). The assumption that "$L_\alpha$ is a set" is equivalent to the existence of a well-founded model of ZF. From a philosophical standpoint, this is a rather strong assertion. For example, it implies that ZF is not only consistent but also $\omega$-consistent and much more...

Martin’s Philosophical Issues About the Hierarchy of Sets

Of course there are no universally agreed-upon answers to these philosophical questions, and if you are interested in Martin's views specifically, then I suggest that you read his articles. Meanwhile, allow me simply to explain a few of the issues arising in the specific questions you mention.

"One cannot quantify over everything." This is a reference to the predicative/impredicative debate in the philosophy of set theory. One of the objections to the replacement and collection axioms is that they are used to describe sets by means of properties of a totality of which they themselves are a member. That is, you define a subset of $B=\{a\in A\mid \varphi(a)\}$, but $\phi(a)$ may be a very complicated property that quantifies over the entire universe, referring to objects and properties of objects, including $B$ itself. But also, it can be a reference to the cumulative view of set theory as building up more and more sets in a process that is never completed, and in this case, it may not be sensible to form sets by means of properties holding in the entire universe, as though it were completed.
"Each of our statements about the universe of sets can from a different perspective be seen as a statement about some $V_\alpha$." The Levy reflection theorem shows that for any assertion $\sigma(x)$, there is an ordinal $\alpha$ such that $\sigma(x)$ is true if and only if it is true in $V_\alpha$, for any $x\in V_\alpha$. That is, $\sigma$ is absolute between $V_\alpha$ and $V$. Going a bit beyond this, consider the theory denoted "$V_\delta\prec V$", which asserts, in the language with a constant for $\delta$, that $\forall x\in V_\delta\, [\varphi(x)\iff \varphi^{V_\delta}(x)]$. This is the scheme asserting that $V_\delta$ is an elementary substructure of the universe. Although some set theorists are surprised to hear it, this scheme is equiconsistent with ZFC, and any model $M$ of set theory can be elementarily embedded into a model of this theory. (This is done by a simple compactness argument; one writes down the theory $V_\delta\prec V$ plus the elementary diagram of $M$, and observes that the reflection theory shows that it is finitely consistent.) Finally, note that in a model of $V_\delta\prec V$, every sentence can be viewed as an assertion about $V_\delta$, rather than about $V$, since they have exactly the same theory.
"Is the first order theory of $V$ determinate?". This question is asking whether there is a fact of the matter in regard to our set-theoretic questions. For example, does it make sense to say that there is ultimately an answer to the question of whether the Continuum Hypothesis is really true? Or whether large cardinals exist? This question is connected in my mind with issues about whether there is a unique structure that we are investigating when we do set theory---the universe $V$ of all sets---or is there instead a multiverse of possibilities? In other words, is there a final truth of the matter in set theory, or is set theory instead something more like geometry, having a plethora of diverse Euclidean and non-Euclidean worlds? In the slides for my talk at the same conference, I explore the multiverse view in detail.
"Are there levels of the hierarchy whose first order theories are indeterminate? What is the lowest such level?" Some set theorists may view questions about the Continuum Hypothesis to be a source of indeterminateness, in the sense that there is no fact of the matter about CH. But CH is a statement expressible in $H_{\omega_2}$, or alternatively in $V_{\omega+2}$. Martin is asking whether we might expect indeterminateness at lower levels. In his talk at the workshop you mention, I recall him saying that he found it unacceptable to think that there would be indeterminateness arising at the level of $V_\omega$, and that arithmetic truth was absolute in some very strong sense.
"There are many examples of proofs of a statement about one level of the hierachy that use principles about a higher level." This is referring to the fact that mathematicians routinely use higher level objects in order to make conclusions about lower level objects. For example, one might use infinite objects (such as automorphisms of field extensions) in order to make conclusions about finite objects, or very large function spaces or ultrafilters in order to make conclusions about a lower level object. Part of Martin's point was the philosophical concern that if there is indeterminism about features of the higher level objects, then they might seem unsuited for this purpose.

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