IF i have T in piano arithmetic and suppose I add a new symbol c
suppose it is T1
T1=T U {c>1,c>1+1,c>1+1+1.......}
Is it true that every finite subset of T1 is consistent and can we show that a structure M |= T1
I have read in a notes that it is by soundness we can confirm that it is consistent
From my googling I understand that if aal the subsets are consistent then T1 is consistent , how can I show that all the subsets of T1 are consistent and the existence of the structure
For any finite subset of the axioms there is one with the most $1$s on the right. Any model of PA will then satisfy PA plus a finite number of these axioms by taking $c$ larger than this number. This shows that every finite subset of $T1$ has a model (assuming PA has a model) and is therefore consistent. The compactness theorem then promises that the whole set of axioms is consistent and has a model, but the usual model of PA doesn't work, so there must be a model that satisfies PA but is not the usual model.