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In the aftermath of the discoveries in foundations of mathematiC's there was surprisingly little effect on mathematics as a whole. If one looks at stan­ dard textbooks in different mathematical disciplines, especially those closer to what is referred to as applied mathematics, there is little trace of those developments outside of mathematical logic and model theory. But it seems fair to say that there is a widespread conviction that the principles embodied in the Zermelo - Fraenkel theory with Choice (ZFC) are a correct description of the set theoretic underpinnings of mathematics. In most textbooks of the kind referred to above, there is, of course, no discussion of these matters, and set theory is assumed informally, although more advanced principles like Choice or sometimes Replacement are often mentioned explicitly. This implicitly fixes a point of view of the mathemat­ ical universe which is at odds with the results in foundations. For example most mathematicians still take it for granted that the real number system is uniquely determined up to isomorphism, which is a correct point of view as long as one does not accept to look at "unnatural" interpretations of the membership relation.

### Introduction

Abstract
For the convenience of the reader, we give a short and informal resume of the content, without going into technical details.
Vladimir Kanovei, Michael Reeken

### 1. Getting started

Abstract
This chapter introduces HST, Hrbaček set theory.
Vladimir Kanovei, Michael Reeken

### 2. Elementary real analysis in the nonstandard universe

Abstract
Our main subject in this Chapter will be the development of nonstandard real analysis in the frameworks of the foundational scheme “$${\text{WF}}\xrightarrow{*}\left| {\left[ {{\text{in}}{\kern 1pt} {\text{H}}} \right]} \right.$$” of HST (as explained in § 1.2a). Of course, by no means can we hope to prove any new mathematical fact this way: indeed, if Φ is an ∈-sentence then Φ wf, the relativization of Φ to WF, is provable in HST if and only if Φ is a theorem of ZFC (Theorem 1.1.14). Yet a broader “external” view brings us new insights into the nature of very common mathematical objects, or rather restores, at the level of full mathematical rigor, mathematical ideas and constructions once successfully employed by the masters of early calculus but then abandoned as too vague to admit rigorous treatment.
Vladimir Kanovei, Michael Reeken

### 3. Theories of internal sets

Abstract
The class ⌷ of all internal sets, or, more exactly, the structure ⌷, is a very important substructure of the nonstandard set universe of HST because it contains many typically nonstandard objects like infinitely large or infinitesimal numbers (see Chapter 2). It will be demonstrated (Theorem 3.1.8) that 〈⌷ ; ∈, st〉 satisfies the axioms of bounded set theory BST a variant of Nelson’s internal set theory IST.
Vladimir Kanovei, Michael Reeken

### 4. Metamathematics of internal theories

Abstract
One of the most important metamathematical issues related to any formal theory is the question of consistency: that is, a theory should not imply a contradiction. As long as minimally reasonable set theories are considered, Gödel’s famous incompleteness theorems make it impossible to prove the consistency in any absolute sense, so that usually the results are given in terms of equiconsistency with some other theory, for instance, ZFC. In this Chapter, we prove that the internal theories IST and BST considered above are equiconsistent with ZFC, that is, consistency of ZFC logically implies consistency of both BST and IST. (For the opposite direction, if IST or BST is consistent then obviously so is ZFC as a subtheory of each of IST, BST see Exercise 3.1.3.)
Vladimir Kanovei, Michael Reeken

### 5. Definable external sets and metamathematics of HST

Abstract
Metamathematical studies of nonstandard theories continue in this Chapter with the aim to prove the main metamathematical properties of HST including its standard core interpretability in ZFC (Theorem 1.1.14) and internal core interpretability in BST (essentially, Theorem 3.1.10) and consequences related to conservativity etc. Section 5.1 introduces all necessary notation and presents the main results (Theorem 5.1.4 and Corollary 5.1.5).
Vladimir Kanovei, Michael Reeken

### 6. Partially saturated universes and the Power Set problem

Abstract
Unlike the model theoretic version of nonstandard analysis, which offers a multiplicity of nonstandard structures with various properties, HST directly provides us with a unique universe | of all internal sets, saturated in a certain maximally possible way, and embedded in the external universe H of all sets. This may appear too boring for a specialist accustomed to deal with peculiar nonstandard models with sometimes hardly achievable properties.
Vladimir Kanovei, Michael Reeken

### 7. Forcing extensions of the nonstandard universe

Abstract
Recall that the class L[I] of sets constructible from internal sets was employed in Chapter 5 to obtain some consistency theorems. For instance Theorem 5.5.8 implies that it is consistent with HST that I-infinite internal sets of different I-cardinalities are necessarily non-equinumerous. It would be in the spirit of mathematical foundations to ask whether the negation of this sentence, that is the existence of equinumerous I-infinite internal sets of different I-cardinalities, is also consistent.
Vladimir Kanovei, Michael Reeken

### 8. Other nonstandard theories

Abstract
The “Hrbaček paradox” (Theorem 1.3.9) can be viewed as the statement of inconsistency of the conjunction of the four following axioms, over a weak nonstandard theory:
• Collection;
• either of the axioms of Choice and Power Set;
• standard size Saturation;
• Standardization.
Vladimir Kanovei, Michael Reeken

### 9. “Hyperfinite” descriptive set theory

Abstract
Descriptive set theory studies those subsets of topological spaces (called pointsets) which can be defined, by means of a list of specified operations including, e.g., complement, countable union and intersection, projection, beginning with open sets of the space. Classical descriptive set theory (DST) considers mainly sets in Polish (that is, separable metric) spaces, this is why we shall identify it here as Polish descriptive set theory.
Vladimir Kanovei, Michael Reeken