caa a22 4500 465793991 CHVBK 20180323112032.0 cr unu---uuuuu 170327e19900601xx s 000 0 eng 10.1007/BF01807624 doi (NATIONALLICENCE)springer-10.1007/BF01807624 Moore Gregory McMaster University, USA aut Proof and the infinite [Elektronische Daten] [Gregory Moore] Conclusion: When modern infinitary logic arose in the mid 1950s, it was motivated primarily by the desire to extend first-order logic to a stronger logic that would retain certain desirable properties of first-order logic. Those who invented this infinitary logic showed little awareness of their predecessors' work using infinitely long formulas, such as that of Löwenheim and Carnap. But, thanks to this modern infinitary logic, the notion of formal proof was enlarged in a fundamental way. During the 20th century the notion of formal proof has been one of the most fertile and important notions in mathematical logic. The distinction between syntactic and semantic notions (for example, proof vs. truth, consistency vs. satisfiability, theorem vs. logical consequence) is something that everyone well educated in mathematics should be aware of. While educators can reasonably differ as to when students should learn these notions, they can hardly deny the fact that students of mathematics should understand formal proof — its uses and its limitations. Infinitary logic is important in overcoming certain of these limitations, and so has a significant place in the education of all those who wish to understand mathematics in depth. The Ontario Institute for Studies in Education, 1990 Interchange Kluwer Academic Publishers 21/2(1990-06-01), 46-60 0826-4805 21:2<46 1990 21 10780 https://doi.org/10.1007/BF01807624 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/BF01807624 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Moore Gregory McMaster University, USA aut NATIONALLICENCE 773 0- Interchange Kluwer Academic Publishers 21/2(1990-06-01), 46-60 0826-4805 21:2<46 1990 21 10780 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer