36:
287:
1, 2, 3, ... the letters of alphabet and some special symbols like "+", "⇒", "(", ")", etc.), give a finite number of propositions expressed in those symbols, which were to be taken as "foundations" (the axioms), and some
234:, such as, "What is the true base of mathematics?" The program was to be able to rewrite all mathematics using an entirely syntactical language
119:
166:
In standard mathematics, an operation is finitary by definition. Therefore, these terms are usually only used in the context of
57:
100:
376:
72:
336:
in the argument will necessarily be finite since the proof is finite, but the number of axioms from which these are
231:
53:
79:
141:
46:
277:
thought is based on a finite number of principles and all the reasonings follow essentially one rule: the
345:
212:
86:
68:
208:
191:
149:
308:) without the need to rely on ingenuity. The hope was to prove that from these axioms and rules
289:
204:
199:
167:
284:
361:
93:
370:
239:
341:
279:
160:
194:(including all assumptions) that can be written on a large enough sheet of paper.
216:
133:
35:
183:
17:
313:
243:
283:. The project was to fix a finite number of symbols (essentially the
312:
the theorems of mathematics could be deduced. That aim is known as
227:
187:
152:
137:
155:, i.e. if it has a finite number of input values. Similarly, an
300:
using only the stated rules (which make mathematics look like a
292:
which would model the way humans make conclusions. From these,
29:
362:
Stanford
Encyclopedia of Philosophy entry on Infinitary Logic
294:
regardless of the semantic interpretation of the symbols
27:
Qualifies an operation with a finite number of arguments
273:
The stress on finiteness came from the idea that human
186:
of symbolic propositions starting from a finite set of
60:. Unsourced material may be challenged and removed.
215:, for instance, as derived from an infinitary
246:), "it does not matter if we call the things
230:in the early 20th century aimed to solve the
8:
203:studies logics that allow infinitely long
120:Learn how and when to remove this message
325:
211:. In such a logic, one can regard the
182:is one which can be translated into a
296:the remaining theorems should follow
7:
58:adding citations to reliable sources
25:
340:is infinite when the system has
34:
45:needs additional citations for
1:
344:, e.g. the axiom schemes of
393:
190:. In other words, it is a
159:operation is one with an
346:propositional calculus
232:problem of foundations
213:existential quantifier
332:The number of axioms
54:improve this article
377:Mathematical logic
290:rules of inference
238:. In the words of
163:of input values.
302:game with symbols
236:without semantics
180:finitary argument
174:Finitary argument
130:
129:
122:
104:
16:(Redirected from
384:
349:
330:
200:infinitary logic
168:infinitary logic
125:
118:
114:
111:
105:
103:
62:
38:
30:
21:
392:
391:
387:
386:
385:
383:
382:
381:
367:
366:
358:
353:
352:
331:
327:
322:
225:
176:
161:infinite number
126:
115:
109:
106:
63:
61:
51:
39:
28:
23:
22:
15:
12:
11:
5:
390:
388:
380:
379:
369:
368:
365:
364:
357:
356:External links
354:
351:
350:
324:
323:
321:
318:
242:(referring to
224:
221:
175:
172:
128:
127:
42:
40:
33:
26:
24:
18:Finitary logic
14:
13:
10:
9:
6:
4:
3:
2:
389:
378:
375:
374:
372:
363:
360:
359:
355:
347:
343:
342:axiom schemes
339:
335:
329:
326:
319:
317:
315:
311:
307:
303:
299:
295:
291:
286:
282:
281:
276:
271:
269:
265:
261:
257:
253:
249:
245:
241:
240:David Hilbert
237:
233:
229:
222:
220:
218:
214:
210:
206:
202:
201:
197:By contrast,
195:
193:
189:
185:
181:
173:
171:
169:
164:
162:
158:
154:
151:
147:
143:
139:
135:
124:
121:
113:
102:
99:
95:
92:
88:
85:
81:
78:
74:
71: –
70:
66:
65:Find sources:
59:
55:
49:
48:
43:This article
41:
37:
32:
31:
19:
337:
333:
328:
309:
305:
304:more than a
301:
297:
293:
280:modus ponens
278:
275:mathematical
274:
272:
267:
263:
259:
255:
251:
247:
235:
226:
198:
196:
179:
177:
165:
156:
145:
131:
116:
107:
97:
90:
83:
76:
64:
52:Please help
47:verification
44:
217:disjunction
134:mathematics
334:referenced
205:statements
184:finite set
157:infinitary
148:if it has
110:April 2012
80:newspapers
69:"Finitary"
256:beer mugs
228:Logicians
142:operation
371:Category
314:logicism
298:formally
285:numerals
244:geometry
146:finitary
306:science
223:History
94:scholar
338:chosen
268:planes
260:points
252:tables
248:chairs
209:proofs
188:axioms
150:finite
96:
89:
82:
75:
67:
320:Notes
264:lines
192:proof
153:arity
140:, an
138:logic
101:JSTOR
87:books
266:and
254:and
207:and
136:and
73:news
310:all
270:."
258:or
144:is
132:In
56:by
373::
316:.
262:,
250:,
219:.
178:A
170:.
348:.
123:)
117:(
112:)
108:(
98:·
91:·
84:·
77:·
50:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
