Below N stands for 1 2 3 Given a firstorder language L and

Below, N_+ stands for {1, 2, 3, ...}. Given a first-order language L and a sentence sigma, the spectrum of a is defined as spec (sigma) = {n N_+: there exists a model M |= sigma such that the universe of M has n elements}. For each set X below find a language L and a sentence sigma in L such that spec (s) = X. Justify your answers (a) X = N_+, (b) X = {n N_+: n is even}, (c) X = {n N_+: n is a power of 2}

Solution

given N_{+={1.2.3.......}}

_{ax specifies universal}

_{ex specifies existential}

A(written in reverse indicates for all)

E(written as mirror reflection indictes there exists)

_{Language l and sentence (spe) are:-}

A)

_Language

_{(ax) x => lies between{1,2,3.....n}}

sentence:

A(written in reverse) N₊        E(there exists written as mirror image) x        such that x lies between {1,2....n}

_{b)Language :}

_{(ax) x => (ex) x where { {1,2,3..n.}/2 = 0 }}

_SENTENCE:

A(written in reverse) N₊        E( written as mirror image) x        such that (N₊/2 )= 0

c)

Language:

(ax) x => (ex) x where x is multiples of (2^n).

^SENTENCE:

A(written in reverse) N₊        E(written as mirror image) x        such that x belongsto (2^{N+) .}