Looking at this today, it seems sort-of obvious. So he created a really clever mechanism for numerical encoding. The Principia logic is minimal and a bit cryptic, but it was built for a specific purpose: to have a minimal syntax, and a complete but minimal set of axioms. The whole idea of the Principia logic is to be purely syntactic. The basic building blocks of the logic are variables.
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Looking at this today, it seems sort-of obvious. So he created a really clever mechanism for numerical encoding. The Principia logic is minimal and a bit cryptic, but it was built for a specific purpose: to have a minimal syntax, and a complete but minimal set of axioms. The whole idea of the Principia logic is to be purely syntactic.
The basic building blocks of the logic are variables. When we think of logic, we usually consider predicates to be a fundamental thing. A predicate is just a shorthand for a set, and a set is represented by a variable. Variables are stratified. Again, it helps to remember where Russell and Whitehead were coming from when they were writing the Principia.
In order to prevent that, they thought that they could create a stratified logic, where on each level, you could only reason about objects from the level below. A first-order predicate would be a second-level object could only reason about first level objects.
A second-order predicate would be a third-level object which could reason about second-level objects. No predicate could ever reason about itself or anything on its on level. This leveling property is a fundamental property built into their logic.
The way the levels work is: Type one variables, which range over simple atomic values, like specific single natural numbers. Type-1 variables are written as ,. Type two variables, which range over sets of atomic values, like sets of natural numbers. A predicate, like IsOdd, about specific natural numbers would be represented as a type-2 variable.
Type-2 variables are written , , … Type three variables range over sets of sets of atomic values. The mappings of a function could be represented as type-3 variables: in set theoretic terms, a function is set of ordered pairs.
A function, in turn, would be represented by a type-4 variable — a set of ordered pairs, which is a set of sets of sets of values. As with variables, the signs are divided into stratified levels: Type-1 signs are variables, and successor expressions. Once you have signs, you can assemble the basic signs into formulae. Everything else from predicate logic can be defined in terms of combinations of these basic formulae. With the syntax of the system set, the next thing we need is the basic axioms of logical inference in the system.
There are five families of axioms. These are defined as axiom schemata, which can be instantiated for any set of formalae ,.
Yes, that is an isomorphism. Whether a big deal or not, we use isomorphisms like this to answer questions about a "hard" domain by mapping the problem into an "easier" domain and working there. We can then map the answer back out if we need to. Exercise: Describe some instances of where this way of working is done. Something to Think About Suppose you had a formal system that was r. Now consider the set of all numbers S that are not encodings of some theorem in the system. Could there exist a computer program that given some number can tell you whether or not it is in S?
Sign up using Email and Password. Mellin transform sin 2x. John Red 2. There are infinitely many prime numbers; the beginning of the sequence is 2, 3, 5, 7, 11, 13, 17, …. So there is no real benefit to encoding the subscripts of the variables at the beginning rather than just using a different number for each variable. Since p is a prime element note that the irreducible element property is usedwe get.