Models
A model of the above example is a left-ended Post-Turing machine tape with its fixed "head" located on the left-end square; the system's relation is equivalent to: "To the left end, tack on a new square □, right-shift the tape, then print ■ on the new square". Another model is the natural numbers as created by the "successor" function. Because the objects in the two systems e.g. ( □, ■□, ■■□, ■■■□ ... ) and (0, 0', 0, 0', ...) can be put into a 1-1 correspondence, the systems are said to be (simply) isomorphic (meaning "same shape"). Yet another isomorphic model is the little sequence of instructions for a counter machine e.g. "Do the following in sequence: (1) Dig a hole. (2) Into the hole, throw a pebble. (3) Go to step 2."
As long as their objects can be placed in one-to-one correspondence ("while preserving the relationships") models can be considered "equivalent" no matter how their objects are generated (e.g. genetically or axiomatically):
- "Any two simply isomorphic systems constitute representations of the same abstract system, which is obtained by abstracting from either of them, i.e. by leaving out of account all relationships and properties except the ones to be considered for the abstract system." (Kleene 1935:25)
Read more about this topic: Object Theory
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