Proof and the infinite

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.