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.