Phenomenology, Logic and the Philosophy of Mathematics Richard Tieszen Index More Information

Cambridge University Press 0521837820 - Phenomenology, Logic and the Philosophy of Mathematics Richard Tieszen Index More information

Index

absolutism, 44, 332 see also Frege, G.; natural numbers; abstraction, 34, 36–37, 78, 84, 325, 327, Peano arithmetic (PA); Poincaré, H.; 328 primitive recursive arithmetic (PRA); formal and material, 28 Weyl, H. see also founding/founded distinction artificial intelligence, 288 absurdity associationism, 210 formal a priori, 27 authentic/inauthentic distinction, 3, 42 material a priori, 27 axiom systems algorithmic methods, 32 definite formal, 4, 11–12, 29 alienation, 42 analysis, paradox of, 320 Becker, O., 8, 9, 14, 62, 83, 126, 237, 247, analytic a priori 254, 260, 268 judgments, 28 Bell, D., 329 see also analyticity; analytic/synthetic Benacerraf, P., 58, 64, 172 distinction Bernays, P., 108, 153, 245 Analytic and Continental philosophy, 1–2, Beth models, 289, 293 44–45, 66 BHK interpretation, 229, 232, 238, 242, analyticity, 185–190 245 rational intuition and, 188–190 biologism, 23 see also Frege, G.; Gödel, K.; Poincaré, Bishop, E., 228, 288 H.; Quine,W.V.O. Bolzano, B., 24, 51, 154, 318 analytic/synthetic distinction, 318 Boolos, G., 187 in Husserl, 27–28 Brentano, F., 1, 22 antinomies, 131 Brouwer, L. E. J., 7, 8, 118, 227, 228, 231, see also paradoxes 234, 235, 248, 249, 253, 254, 266, 278, antireductionism 283, 296 in Husserl, 33 see also intuitionism see also reductionism apophantic analytics, 28–29 calculation Aristotelian realism, 127 in science, 36, 40, 41–42 Aristotle, 127 Cantor, G., 9, 86, 126, 285 arithmetic Carnap, R., 1, 9, 70, 178, 179, 182 in Husserl, 2–3, 22, 32 Carnap’s program, 136–138, 182 Husserlian transcendental view of, 319, Cartesian dualism, 224 325–336 Cayley, A., 77

349

350 Index

Chihara, C., 211 substitutions of, 182–183, 193–194 choice, axiom of, 99, 196 see also Gödel, K.; intentionality choice sequences, 229 Continental philosophy, 21, 36, 100 see also Weyl, H. continuum Church’s Thesis, 217, 230, 231, 288 atomistic, 265, 266 complexity intuitive, 229, 263–271 algorithmic, 157 Weyl and the, 263–271 concealment, 45 continuum hypothesis (CH), 98, 99–100, concepts, 127, 149–176 139, 144, 167, 187–188, 196 are not subjective ideas, 170–171 conventionalism, 50, 66, 81, 136, 166 as intensional entities, 157 countersense (Widersinn), 27, 28 and intentionality, 156 Couturat, L., 296 and particulars, 223 Curry, H., 312 properties of, 169 concrete mathematics, 150–152 Da Silva, J., 262 consequence logic (Konsequenzlogik), 28 Davidson, D., 224 consistency decidability, 158, 161, 274 and existence, 30 see also Gödel, K.; incompleteness in geometry, 82 theorems see also Gödel, K.; incompleteness Dedekind, R., 259, 284 proofs Dennett, D., 199 consistency proofs, 64, 286 Derrida, J., 21 forarithmetic, 133–135, 152, 221, Descartes, R., 25 222 Detlefsen, M. in geometry, 31, 87 on Poincaré, 301–303 of ZF + CH, 146, 152 dual aspect theories, 224 see also Gödel, K.; incompleteness Dummett, M., 14, 192, 235, 277, 290, 329 proofs critics of on intuitionism, 233 constitution and extensional view of meaning, 244 of idealities, 4, 13, 26 and intuitionism, 227–247 of natural numbers, 327–329 meaning-theoretic argument for of space, 31 intuitionism, 231–233 constructibility, axiom of, 109, 196, 197 constructions, 228, 230, 331 eidetic reduction, 69, 95 as full manifestations of knowledge of see also free variation meaning, 233–237 empiricism, 197, 333 in intuitionism as fulfilled intentions, Husserl’s critique of, 23 237–247, 254 essences, 127–130 and mechanical (Turing) computation, formal and material, 128 230, 237, 240 in geometry, 69–88 see also intuitionism; proof hierarchies of geometric, 76–80 constructive recursive mathematics as invariants, 71, 74 (CRM), 228, 231 and meaning, 12–13 constructivism, 14, 61–62, 107, 227–247, morphological, 31, 69, 84 276–293 reductionism about, 130 and classical mathematics, 273–275, and sets, 76 284–286 Euclid, 316 and weak counterexamples, 284 Euclidean geometry in Weyl, 248–275 see geometry see also intuitionism evidence, 30, 61, 96, 103, 245, 252, 332, 333 content and proof, 277–293 intentional, 52–53, 96, 199, 210, 279, excluded middle, principle of, 229, 238, 280, 327 269, 284 mathematical, 190–199, 318 as an idealization, 284, 301, 332

Index 351

existentialists, 100 notion of sense (Sinn), 139, 154, 298, extensions 321, 330 determined by intentions, 243–245, on psychologism, 316, 318 280–281 and realism (platonism), 316 see also Dummett, M.; Frege, G.; Quine, and role of intuition in arithmetic, 316, W. V. O.; Weyl, H. 324 extensionality, axiom of, 99 telescope analogy in, 171 on thoughts and judgments, 329–330 Feferman, S., 111, 217, 249, 266 Freudenthal, H., 239 Fichte, G., 248, 250–251, 253, 261, 272, fulfillment, 3, 60, 252 273 and intuitionistic constructions, see fictionalism, 48, 64–65, 285 constructions Field, H., 48, 212 and mathematical intentions, see proof finitism, 63 see also evidence; intentionality and intuitionism, 233, 234 functions Folina, J., 306, 307 as rules or graphs, 230 Føllesdal, D., 97, 183 formalism, 66, 230, 295, 310–313 Gadamer, H-G., 44 in Husserl, 3, 298 Galileo, 39, 42 see also Hilbertian formalism genetic phenomenology, 42–44 formalization, 298, 312, 333, 336 see also origins role of in science, 40 Gentzen, G., 232, 290 formalization/generalization distinction, genus/species relations, 28 28, 181, 191, 192 geometry, 31–32, 43, 69–88 formal systems, 4–6, 230, 234 non-Euclidean, 79 see also Hilbertian formalism projective, 77, 79 founding/founded distinction, 14, transcendental, 88 35, 54–55, 159, 258, 259, 304–305, see also space 330 George, A., 234, 235 in arithmetic, 327–329, 330 Gödel, K., 51, 214, 258, 273, 289, 308 in geometry, 83–88 on absolutely unsolvable problems, 119, see also origins 124, 145, 151, 161, 167, 186 free variation (in imagination), 43, on analyticity, 117, 120, 123, 139, 185, 66, 69–89, 98–100, 165–167, 274, 186–190 334 on Aristotelian realism, 120, 176 Frege, G., 1, 9, 14, 25, 51, 60, 126, 147, on Carnap, 120–121, 179 165, 208, 209, 214, 238, 242, 243, on the concept of proof, 114, 116 283, 310 and concepts, 97–100, 154–155, analyticity in, 296, 318 180–181, 185 and the definition of number, 317, 318, concern for constructivity, 113 319, 321 on constructivity and impredicativity, and extensionality and intensionality in 306, 307 arithmetic, 317, 320–321, 322 and content, 120, 138, 139, 178, and gapless formal proofs, 316 179–181, 186, 190–199 on geometric foundations of arithmetic, on conventionalism, 119, 174 322–325 on empiricism, 120, 175–176 on Hilbertian formalism, 312 on Hilbert’s program, 114–115, 121, and Husserl on logic and meanings, 131–132, 138 297–299, 300 and Husserl, 10–13, 93–107, 121–122, and Husserl on number, 14–16, 67, 299, 123, 125–148, 149–176, 334 314–336 on the incompleteness theorems, laterview of number, 322–325 113–115, 118–122, 186–187 and logicism, 295, 296, 298, 314–322, and inexhaustibility of mathematics, 333, 334 139, 144, 173, 197

352 Index

Gödel, K. (cont.) Hilbert’s program, 146, 150, 182, 220, and the intuition of concepts, 149–176, 273 178, 189 see also Gödel, K. on mathematical intuition, 101–102, Hintikka, J., 105 138, 140 historical a priori, 42–44 and meaning clarification, 141–142, 153, historicism, 23, 43–44, 328 154, 155, 177, 178–180, 181–183, 184, holism, 233 223 in Quine, see Quine, W. V. O. on minds and machines, 106, 110, 114, scope of, 183, 184, 195 145, 153, 173–174 horizons, 160 on nominalism, 120, 174 of acts, 63, 99, 158, 256, 268, 298, 334 philosophical writings on logic and Hume, D., 185 mathematics, 112–124 Husserl, E. and platonism, 116–118, 119 stages of his work, 2–4, 6–7 on positivism, 175 hyletic data, 38 on psychologism, 120, 175 and Quine, 177–200 idealism, 249–255, 270 and rational intuition, 122, 138–140, subjective, 62 144, 179, 188–190, 192 transcendental, 4, 248, 249, 253 realism and idealism in, 101, 105–107 idealization(s), 31, 39, 63, 160 on Russell, 116–117 classical real numbers as, 265 on set theory, 118, 139–140, 144–145, in geometry, 84–85 187, 286 in intuitionism, 245 and Turing, 115 principle of excluded middle as, 284, Goldfarb, W., 296 301, 332 Grassmann, H., 31, 70 ideal objects, 2, 3, 25–26, 56, 57–58, 168, Griss, G. F. C., 239 169, 325–327, 330 group theory, 76 bound, 57 free, 57 Habermas, J., 21 individual, 168 Hahn, H., 136 as omnitemporal, 25 Hallet, M., 307 universal, 168 Hebb, D., 204, 210 ideation, 69 Heidegger, M., 1, 21, 23, 44, 83, 130 see also free variation Heinzmann, G., 303 illusion, 60, 104, 139, 326 Helmholtz, H., 70 imagination, 72–73 hermeneutics, 44 see also free variation Heyting, A., 8, 14, 62, 126, 227, 228, 231, impredicativity, 258, 305–310 234, 237 and limits of constructive intuition, on degrees of evidence in intuitionism, 306–308 245 see also vicious circle principle and Freudenthal correspondence, inauthentic presentations 238–239 see authentic/inauthentic distinction on intuitionistic constructions, 237–239, incompleteness theorems, 6, 32, 131–138, 242, 247, 254, 279, 287 151, 153, 173, 179, 192–194, 216–224, Hilbert, D., 106, 126, 135, 153, 178, 179, 230, 273, 276, 314, 333 249, 253, 271–273, 284, 312 and Husserl’s definite axiom systems, optimism in the work of, 135–136 11–12 Hilbertian formalism, 48–49, 173, 295, 301, see also Gödel, K.; Penrose, R. 311, 315, 333 infinity and Husserl, 9 actual, 305 and immediate intuition, 151, 153 in intuitionism, 229 see also Frege, G.; Gödel, K.; Poincaré, see also set; set theory H.; Weyl, H. infinity, axioms of, 123, 139, 144, 167, 187

Index 353

informal rigor, 49–50, 66, 135, 143, 198, Kant, I., 25, 57, 62, 65, 88, 95, 101, 107, 298, 334 122, 145, 224, 228, 233, 235, 246, 248, instrumentalism, 42 249–250, 253, 283, 284, 297, 300, 305, intensions 318, 325 see concepts; Frege, G., meaning, theory on intuition and concepts, 250, 281 of; meanings; Weyl, H. Kaufmann, F., 8 intentionality, 9, 37, 51–54, 95–97, 126, Kim, J., 59 147, 150, 155–157, 161–162, 171–172, Kleene, S., 216, 287, 289 174, 180, 188, 189, 199, 211, 221, 223, Klein, F., 70, 77 224, 227, 237, 241, 242, 252, 261, Erlanger program, 76 279–293, 294, 297, 303, 311, 321, knowledge 327 causal theory of, 58 derived, 240, 288 of mathematical objects, 58–65 group, 235, 236 and proof, 276–293 intentional stance, 171 see also evidence; intuition intentions Kolmogorov, A. N., 239 and intuitionistic mathematics, 237–247 interpretation of intuitionistic logic, 287 see also intentionality Kreisel, G., 49, 93, 97, 101, 109, 113, 334 intersubjectivity, see solipsism Kripke models, 289, 291–293 intuition, 29, 32, 61–62, 96, 98, 102, 252, 300 least upperbound principle, 257, 263 analogies between sensory and Leibniz, G., 24, 25, 80, 89, 93, 95, 129, 154 categorial, 75 Leibnizian preestablished harmony, 224 in arithmetic, 329–335 Lie, S., 70 categorial, 3, 128, 138 lifeworld (Lebenswelt), 35–36, 38, 85, 305 as distinct from conjecture, 75 lived body, 37, 83 of essences, 12, 69–89, 98, 128, 129, logic 273 intensional, 3, 162, 242, 280, 299, 317, and impredicativity, 306–308 320–321 in intuitionism, 228, 233, 246 of noncontradiction in Husserl, 27–29 mathematical, 7, 63–64 as science of all possible sciences, 24–25, of particular mathematical objects, 61 318 and vicious circularity, 257–258 as a theory of reason, 24–26 of sets, 76 threefold stratification of in Husserl, see also evidence; Frege, G.; Gödel, K.; 26–30, 82 Maddy, P.; Poincaré, H.; Weyl, H. transcendental, 35 intuitionism, 62, 146, 182, 227–247 of truth in Husserl, 29–30, 332 and Husserl, 7–8 logicism, 230, 295–305 in relation to classical mathematics, see also Frege, G.; Poincaré, H. 244–246 Lotze, H., 24 see also Brouwer,L.E.J.;constructivism; Lucas, J. R., 215, 217, 220, 221 Dummett, M.; Heyting, A. intuitive/conceptual distinction, 7 Maddy, P., 59, 108 see also Kant, I. and the analogy between perception of intuitive/symbolic distinction, 3, 33, 254, sets and perception of physical 271–275 objects, 206–208 see also authentic/inauthentic on Benacerraf, 204, 208, 209 distinction on compromise platonism, 201–214 invariants, 55–58, 71–75, 76–83, 88–89, monistic and dualistic, 211–213 166, 169, 170, 171–172, 206, 326 on GödelianPlatonism, 201–202, 203 levels of, 74 on intrinsic and extrinsic justification, 202, 211 Jackson, F., 235, 241 on the justification of set-theoretic judgments, 26–30, 34 axioms, 209–211

354 Index

Maddy, P. (cont.) Mohanty, J. N., 321 and mathematical intuition, 204–208, multiple reduction problem, 299, 322 210, 211 on natural numbers, 208–209 Nagel, T., 36, 124, 224 on Quine/Putnam platonism, 201, naturalism, 11, 23, 38, 206, 213–214 202–203 natural numbers on sets of physical objects, 204–206 concept of, 160 Mahnke, D., 9 and lawlike sequences, 229 manifolds, 29 set-theoretic treatment of, 258 definite, 4, 11–12, 29 see also arithmetic Euclidean, 85 natural standpoint, 95–97 manifolds, theory of noema, 96, 280, 327 (Mannigfaltigkeitslehre), 31, 40, 82, nominalism, 47–48, 56, 57, 58, 136 86 nonsense (Unsinn), 27 in modern geometry, 86 numbers Markov, A. A., 228 natural, see arithmetic; natural numbers Markov’s principle, 231 real, see real analysis Martin-Löf,P., 62, 278–279, 287, 289–291 mathematical objects, 55–65 ontologies see also ideal objects; invariants spatial, 80–83 mathematization of nature, 39–42, 168 ontology mathesis universalis, 10, 12, 25 of concepts, 167–169 meaning formal, 4, 28, 29, 86, 298 in classical mathematics, 62 regional, 4, 28, 29, 33, 86, 181, 298 constructive, 229, 242, 245–246, 247 origins determined by use, 232, 244 forgetting, 42, 310 and essence, 12–13 of geometry, 31, 83–88 games-meaning, 4–6, 9, 41 of mathematical content, 54, 66, 334, in geometric formulas, 82 335 see also Dummett, M.; Frege, G.; Gödel, of natural number concept, 2, 15, 16, K.; Poincaré, H.; Quine,W.V.O.; 325–329 Weyl, H. see also founding/founded distinction meaning, theory of intensional, 244 Pagels, H., 93 and intentionality, 53–54, 297–299 paradoxes, 262, 305 and intuitionism, 227–247 Grelling, 258 see also Frege, G.; intentionality Russell, 258, 310, 317, 322 meaning clarification, 6, 12–13, 65–67, set-theoretic, 104 150, 163–165, 166–167, 274 Paris-Harrington theorem, 143, 187 see also Gödel, K. Parsons, C., 49, 94, 109, 177, 190–192, 194, meaning-fulfillment, 29 196, 203, 246, 306 and Weyl, 261–263 parts and wholes, theory of, 27, 267 see also evidence; intentionality; intuition Peano arithmetic (PA), 132, 133, 151, 192, meaning-intention, 30 220, 221, 334 and Weyl, 261–263 Peirce, C. S., 303 see also intentionality Penrose, R., 215–224 meanings compared with Gödel’s anti-mechanism, ideal, 6, 25–26 219–222 mechanically computable function, 143 on noncomputational processes in see also Turing neuroscience, 223 Merleau-Ponty, M., 21, 83, 100 on the incompleteness theorems, minds Mill, J. S., 131, 184, 318, 335 and machines, 216–224 minds and machines, 135, 312 and Penrose-Turing theorem, 216 see also Gödel, K.; Penrose, R. and platonism, 219, 224

Index 355

perception primitive recursive arithmetic (PRA), 132, Husserl’s analysis of, 102–104 133, 151 phenomenological method (epoche´ ), 95, proof, 14, 114, 301 98, 147, 164 automated, 288 and set theory, 98–100 canonical and noncanonical phenomenological reduction, 95, 96, 97, constructive, 290 103, 104, 183, 251 classical and constructive, 284–286 phenomenology concept of, 161 as eidetic science, 4, 37 epistemological conception of, 277–293 as science of all sciences, 38 and evidence, 277–293 philosophy as rigorous science, 38, 94, 100, formalist conception of, 276 129, 130, 143 and fulfillable intentions, 276–293 philosophy of mind, 9 indirect, 285 and intuitionism, 227–247 and intentionality, 279–293 Piaget, J., 209 and intuitionism, 276, 278 Plato, 93 and Kripke models of intuitionistic logic, platonism, 4, 46, 163, 310, 316, 335 291–293 Dummett’s view of, 232 in Martin-Löf’s work, 289–291 in Gödeland Husserl, 13 and ontological platonism, 276 in Husserl, 26, 57–58, 74–75, 147 and programs that satisfy specifications, intuitionistic view of, 229 287–288 phenomenological vs. classical and realizations of expectations, 286–287 metaphysical, 169, 327 and solipsism, 283–284 see also Frege, G.; Gödel, K.; Maddy, P.; and solutions of problems (in Penrose, R.; Quine,W.V.O. Kolmogorov), 287 Poincaré, H., 14, 118, 249, 253, 259, and Turing machine computations, 288 294–313 see also Gödel, K.; Weyl, H. and analyticity, 296 psychologism, 2, 23, 25, 56, 147, 283, 316, on the Cantorians, 297, 304, 305 319 and formalism, 310–313 on Hilbert, 311 Quine, W. V. O., 49, 146, 233, 235, 329 and Husserl, 294–313 on analyticity, 184, 185, 189 and impredicativity, 295, 297, 305–310 and behaviorism, 199 on intuition, 297, 300 empiricism in, 183 and logic, 297 extensionalism in, 190 and logicism, 295–305 and Gödel, 177–200 on mathematical induction, 296, 304 holism in, 183–185, 190, 193–194, 195, and meaning theory, 299–300, 309–310 196 on natural numbers, 15, 258, 294, 295, indispensability argument, 197, 198 297, 299, 305 on intentionality, 199 and Zermelo, 308–309 and mathematical content, 200 positivism, 11, 136, 164, 183, 185 mathematics and theoretical hypotheses Husserl’s critique of, 23, 39 of natural science, 190–194 pragmatism, 49–50, 66 on necessity, 184, 191 Prawitz, D., 232, 241, 290 platonism in, 191 predicativism, 146, 228 and set theory, 193, 194–195, 196–198 see also impredicativity skepticism in, 183 prejudices and unapplied parts of mathematics, in science, 37, 44, 142 195–198 prereflective experience, 158, 159 see also reflection rationalistic optimism, 100, 106–107, 124 presuppositionlessness rationalism, 169, 176 as an ideal in philosophy, 44 in Husserl, 21, 34, 333, 335 primary/secondary qualities, 36 see also reason

356 Index

real analysis Searle, J., 235, 237, 240 intuitionistic, 229, 288 separation, axiom of, 308 see also Weyl, H. set realism general concept of, 161, 181 see platonism iterative concept of, 99, 101, 108–109, real numbers 187, 286 see real analysis logical conception of, 108–109 real objects, 3, 25–26, 56, 169 set theory, 80 individual, 168 impredicative, 244, 286, 308, 310 universal, 168 see also Gödel, K.; Maddy, P.; Quine, reason W. V. O.; Zermelo-Fraenkel set and logic, 24–26 theory skepticism about, 39 Shapiro, S., 212 superficialization of, 23 Skolem, T., 142 see also rationalism solipsism, 236, 237, 283–284 reductionism, 56, 181–183, 299 space, 250 and nominalism, 47–48 affine, 78 see also antireductionism elliptical, 78 reference hyperbolic, 78 see extensions; intentionality projective, 78 reflection, 84, 158, 159–160, 161, 303, 327 see geometry on noemata, 98 specious present, 264 see also prereflective experience Spinoza, B., 224 relativism, 23, 25, 44, 206, 208 Steiner, M., reliabilism, 59, 63 Strawson, P. F., 224 replacement, axiom of, 99 subjectivity representational character/relation, and objectivity, 8, 9, 36–37, 50–51, 295, 102–106, 110 318, 327 see also intentionality Sylvester, J. J., 77 Resnik, M., 212 synthetic a priori retention/protention structure, 264 judgments, 27–28 revelation, 45 see also analytic/synthetic distinction Ricoeur, P., 21 Riemann, B., 31, 70, 81 Tarski, A., 219 Rorty, R., 183 technization in science, 40–42 Rota, G.-C., 93, 97 time, 84, 237, 250, 253, 255, 256, 264–265, Russell, B., 9, 116, 238, 249, 253, 260, 296, 270 305, 306 topology, 77, 78–80 transcendental ego, 251 Schlick, M., 1, 136 transfinite Schröder, E., 126 sets, 109 science see also infinity; set theory of all possible sciences, 34 transformations, groups of, 76 see also philosophy as rigorous science Troelstra, A., 50, 233, 238 sciences truth crisis of, 23, 39–44 within horizons, 63, 332 descriptive, 33 Tarski’s theory of, 58, 64 eidetic and empirical, 33, 95, 97, 129 truth logic (Wahrheitslogik), 29–30, 332 exact and inexact, 33 Turing Husserl’s criticism of modern, 22–23 computability, 116, 216, 228, 230, 237, material and formal, 31, 33, 85; see also 288 ontology: formal; ontology: regional machine, 60, 231 types of, 33–34 test, 241 scientism, 10, 36, 45 Turing, A., 98, 110, 115, 143, 174

Index 357

ultrafinitism, 63 and Hilbert’s formalism, 263, 271–273 universal grammar, 26–27 and idealism, 249–255 on intension and extension, 261 van Dalen, D., 50, 233, 235, 249, 269 and meaning in mathematics, 261–263 vicious circle principle, 116, 258, 306 and intuition, 254–261, 268 see also Gödel, K.; impredicativity; and the intuitive/symbolic distinction, Poincaré, H.; Weyl, H. 254, 271–275 vicious circularity, 294, 297 on lawlike and choice sequences, 257, 261, 263–271 Wang, H., 12, 93, 94, 97, 98, 113, 134, 155, and na¨ıveabsolute realism, 251, 271 181 on the principle of excluded middle, on Quine, 198 257, 269 Weierstrass, K., 126 on real analysis, 257, 259–261, 263–271 Weyl, H., 1, 8, 14, 62, 70, 89, 118, 126, and time consciousness, 263–271 248–275, 305 and vicious circularity, 257–258, and Brouwer on the continuum, 262 267–270 Whitehead, A. N., 238 on concept of natural number, 255, 256, Wittgenstein, L., 136, 228, 232, 236, 239, 258, 259 246, 277 on the continuum, 263–271 Wright, C., 234 on empiricism, 253 and extensionally definite concepts Zermelo, E., 49, 126, 307, 308 (categories), 256, 260, 261, 262, 270 Zermelo-Fraenkel set theory (ZF), 13, 151, on formalism, 253–254 152, 161, 196, 222, 304, 309 on formal proof, 254 constructive, 291 and founding/founded distinction, 259 second-order, 188