拙作所關注者為固定點 \ (fixpoints, fixed points) 之數學建構及其建構方式在構造可滿足形式化之素樸集合論 \ (na\"{i}ve set theories) 之語意論解釋\ (semantic interpretations)，即「模型」(models)。同時，為「羅素悖論」(Russell’s Paradox) 提供解方。拙作將探討兩種類型之模型建構方式。其一為「克里普奇式」(Kripke style)；其二為「史考特式」(Scott style)。

本論文對序格理論 \ (order-lattice theory) 中之「固定點定理」(fixpoint theorem) 有完整之討論與證明。而其中之「完備偏序固定點定理」(CPO fixpoint theorem) 及其建構性版本乃克里普奇式模型建構與史考特式模型建構之共同數學基礎。克里普奇式模型建構將伴隨其在真理論中之發展以呈現之。素樸集合論之克里普奇式模型之存在證明與建構，則以真理論版本為基礎加以修改、調整而得。本論文亦提供範疇論版本之固定點定理證明。並以史考特之\ $D_\infty$ 建構一素樸集合論之史考特式模型之實例\ $D_\infty$。最後，將比較兩種建構方式之差異，並對史考特式模型中某類集合概念之定義略作探討。

Fixpoint models provide solutions to liar type semantic paradoxes and set theoretic paradoxes. Liar paradox and Russell's paradox are the most well-known of two groups respectively. Such solutions also establish semantics for na\"{i}ve truth theories and na\"{i}ve set theories. For na\"{i}ve set theories, there are two different approaches to construct such models, \textit{Kripke style construction} and \textit{Scott style construction}.

This thesis focuses on how these models are constructed in different styles. A self-contain presentation of order-lattice theoretic fixpoint theory is given first. Fixpoints are guaranteed to exist and constructed by order-preserving maps on CPO. Then, models are constructed by Kripke style both for na\"{i}ve truth theories and na\"{i}ve set theories. The application of order-theoretic fixpoint theorems is direct in Kripke style. The fixpoint in Scott style construction are given by a category-theoretic account, though it is ZF-set-theoretic object. The connection from order-theoretic fixpoint theorems to category-theoretic version is set up for Scott style construction. An example of Scott style models is bolted by converting Scott's $D_\infty$ for $\lambda$-calculus. A proposal of characterizing singletons in certain Scott style models has a short discussion.

Table of Contents

摘要 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Order-Lattice Theoretic Fixpoint Theorems . . . . . . . . . . . . . . . . . . . . . 4
2.1 Partially Ordered Sets and Lattices . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 CPOs and CPO Fixpoint Theorem . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Scott Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3 Naïve Set Theory and Logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.1 Cantor-Frege Abstraction and Russell’s Paradox . . . . . . . . . . . . . . . . . 22
3.2 Curing Antinomy Classically . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.3 Being Naïve Non-classically . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.4 Notes on CFA, Extensionality and Equality . . . . . . . . . . . . . . . . . . . 24
3.5 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.6 Formal Language and Its Syntax . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.7 Semantics and Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4 Kripke Style Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.1 Semantic Paradox, Tarski’s and Kripke’s Theory of Truth . . . . . . . . . . . . 34
4.2 Fixpoint Model for Kripke’s Truth Theory . . . . . . . . . . . . . . . . . . . . 38
4.3 Fixpoint Model for Naïve Set Theory by Kripke Style Constuction . . . . . . . 43
5 Scott Style Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5.1 Scott’s D∞ for λ-Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
xvii
5.2 Fixpoint Theorem in a Bit of Category Theory . . . . . . . . . . . . . . . . . . 50
5.3 S∞ constructed from D∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
6 Comparison and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
6.1 Comparison of Two Styles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
6.2 Observations in Restall’s suggestions . . . . . . . . . . . . . . . . . . . . . . . 65
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

[1] S. Kripke, “Outline of a theory of truth,” The Journal of Philosophy, vol. 72, no. 19, pp.
690–716, 1975.
[2] P. C. Gilmore, “The consistency of partial set theory without extensionality,” in Axiomatic
Set Theory, ser. Proceedings of Symposia in Pure Mathematics, vol. 13. American Math-
ematical Society, 1974, pp. 147–153.
[3] A. Gupta and N. Belnap, The Revision Theory of Truth. MIT Press, 1993.
[4] G. Restall, “Fixed point models for theories of properties and classes,” Australasian Jour-
nal of Logic, vol. 14, no. 1, pp. 226–245, 2017, 8.
[5] B. A. Davey and H. A. Priestley, Introduction to Lattices and Order, 2nd ed. Cambridge
University Press, 2002.
[6] S. Roman, Lattices and Ordered Sets. Springer, 2008.
[7] M. Fitting, “Notes on the mathematical aspects of kripke’s theory of truth,” Notre Dame
Journal of Formal Logic, vol. 27, no. 1, pp. 75–88, 1986.
[8] A. Visser, “Semantics and the liar paradox,” in Handbook of Philosophical Logic, 2nd ed.,
D. M. Gabbay and F. Guenthner, Eds. Springer, 2004, vol. 11, pp. 149–240.
[9] P. Suppes, Axiomatic Set Theory. Dover Publications, 1972.
[10] H. B. Enderton, Elements of Set Theory. Academic Press, 1977.
[11] T. Libert, “Semantics for naive set theory in many-valued logics: Technique and historical
account,” in The Age of Alternative Logics: Assessing Philosophy of Logic and Mathemat-
ics Today. Springer, 2006, ch. 9, pp. 121–136.
[12] R. T. Brady and R. Routley, “The non-triviality of extensional dialectical set theory,” Para-
consistent Logic: Essays on the Inconsistent, pp. 415–436, 1989.
69
[13] R. Hinnion and T. Libert, “Positive abstraction and extensionality,” The Journal of Sym-
bolic Logic, vol. 68, no. 3, pp. 828–836, 2003.
[14] H. B. Enderton, A Mathematical Introduction to Logic, 2nd ed. Academic Press, 2001.
[15] R. M. Smullyan, Gödel’s Incompleteness Theorems. Oxford University Press, 1992.
[16] H. Field, Saving Truth from Paradox. Oxford University Press, 2008.
[17] S. Blame, “Partial logic,” in Handbook of Philosophical Logic, 2nd ed., D. M. Gabbay and
F. Guenthner, Eds. Springer, 2002, vol. 05, pp. 261–353.
[18] S. Feferman, “Toward useful type-free theories, i,” The Journal of Symbolic Logic, vol. 49,
no. 1, p. 75–111, 1984.
[19] V. McGee, Truth, Vagueness, and Paradox. Indianapolis: Hackett Publishing Company,
1991.
[20] S. Abramsky and A. Jung, “Domain theory,” in Handbook of Logic in Computer Science.
Clarendon Press, 1994.
[21] P. Apostoli, A. Kanda, and L. Polkowski, “First steps towards computably-infinite informa-
tion systems,” in Transactions on Rough Sets II, ser. Lecture Notes in Computer Science,
vol. 3135. Springer, 2005, pp. 151–188.
[22] T. Libert, “Models for a paraconsistent set theory,” Journal of Applied Logic, vol. 3, no. 1,
pp. 15–41, 2005.
[23] S. Awodey, Category Theory, 2nd ed. Oxford University Press, 2010.
[24] H. Simmons, An Introduction to Category Theory. Cambridge University Press, 2011.
[25] T. Leinster, Basic Category Theory. Cambridge University Press, 2014.
[26] J. R. Hindley and J. P. Seldin, Lambda-Calculus and Combinators, an Introduction. Cam-
bridge University Press, 2008.
70
[27] J. Alama and J. Korbmacher, “The Lambda Calculus,” in The Stanford Encyclopedia of
Philosophy, Fall 2023 ed., E. N. Zalta and U. Nodelman, Eds. Metaphysics Research
Lab, Stanford University, 2023.
[28] D. Lewis, Parts of Classes, 1st ed. Wiley-Blackwell, 1991.
[29] S. Mackereth, “Fixed-point posets in theories of truth,” Journal of Philosophical Logic,
vol. 48, pp. 189–203, 2019.