Implicit in the technique of forcing is the following relative consistency result:

If $\mathfrak M$$\vDash$$T$, and therefore $T$ is consistent (where $\mathfrak M$ is the ground model) then if $\mathfrak M$$[$$G$$]$$\vDash$$T^{'}$, $T^{'}$ is consistent (since I am assuming $T$ and $T^{'}$ are first-order theories, this seems an easy consequence of the Goedel completeness theorem).

Also, it is known that class forcings do not always preserve the axioms of $\mathfrak M$$\vDash$$T$ in $\mathfrak M$$[$ $G$ $]$ $\vDash$ $T^{'}$.

The theory $T$ I am specifically interested in is $ZF$ $-$ *Infinity*, that is, $ZF$ with the Axiom of Infinity dropped.

Question: Is there a class forcing extension of $\mathfrak M$$\vDash$ $ZF$ $-$ *Infinity* such that $\mathfrak M$$[$ $G$ $]$$\vDash$$ZF$ $-$ *Infinity* $+$ *Infinity* (or in the alternative, where $\mathfrak M$$\vDash$$ZF$$-$ *Infinity* $+$ $\lnot$*Infinity* , is there a class forcing producing a forcing extension $\mathfrak M$$[$ $G$ $]$ in which $\lnot$*Infinity* fails)? If there is no such class forcing, show why there cannot be such.

Infinitywas independent of $ZF$ $-$Infinity($ZF$ $-$Infinityshould be able to be proven consistent in $PRA$ $+$ $TI({\epsilon_0})$), but Noah's answer suggests that such a project is doomed to fail. $\endgroup$ – Thomas Benjamin Jan 10 '17 at 7:17