Download A Mathematical Prelude to the Philosophy of Mathematics by Stephen Pollard PDF

By Stephen Pollard

This e-book is predicated on premises: one can't comprehend philosophy of arithmetic with no realizing arithmetic and one can't comprehend arithmetic with out doing arithmetic. It attracts readers into philosophy of arithmetic by way of having them do arithmetic. It bargains 298 routines, overlaying philosophically vital fabric, awarded in a philosophically knowledgeable manner. The workouts supply readers possibilities to recreate a few arithmetic that might remove darkness from very important readings in philosophy of arithmetic. subject matters contain primitive recursive mathematics, Peano mathematics, Gödel's theorems, interpretability, the hierarchy of units, Frege mathematics and intuitionist sentential good judgment. The ebook is meant for readers who comprehend easy homes of the normal and actual numbers and feature a few history in formal logic.

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The identity symbol: ‘=’. Three function symbols: ‘S’ (“successor”), ‘+’ (“plus”), ‘·’ (“times”). One proper name: ‘0’ (“zero”). Infinitely many variables: ‘w’, ‘x’, ‘y’, ‘z’, ‘w1 ’, ‘x1 ’, ‘y1 ’, ‘z 1 ’, … Two parentheses: ‘(’, ‘)’. In everyday English, certain expressions refer to individual things or can so refer in appropriate contexts. Examples include proper names (‘Kurt Gödel’), pronouns (‘him’), and descriptions (‘the second son of Marianne Gödel’). The terms of PA are the expressions that play this role in our formal language.

In an interpretation of PA, we (1) specify the range of our bound variables, (2) assign an object from that range to ‘0’, and (3) assign operations defined on that range to each of ‘S’, ‘+’, and ‘·’. If we are not feeling too adventurous we might (1) let our bound variables range over the natural numbers (so that we read ‘⇐x’ as “for all natural numbers x”), (2) let ‘0’ be our name for zero, and (3) assign the operations of immediate succession, addition, and multiplication to ‘S’, ‘+’, and ‘·’.

13 shows: if PA allowed us to say that no natural number codes a PA-proof of ‘0 = 0’, PA would not allow us to prove this unless it allowed us to prove everything. So we will not be able to use a PA-proof to demonstrate the consistency of PA. More generally, we will not be able to prove the consistency of PA using methods formalizable in PA. Note that a PA-proof of PA’s consistency would not be as pointless as it might first appear. The PA-proof would use only finitely many axioms of PA: we would be relying on only finitely many axioms to show that no combination of the infinitely many PA-axioms proves an absurdity.

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