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corrected vagueness in question
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Thomas Benjamin
Thomas Benjamin
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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 forif $\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.

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

If $\mathfrak M$$\vDash$$T$ is consistent (where $\mathfrak M$ is the ground model) then for $\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.

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.

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Thomas Benjamin
Thomas Benjamin
  • 6.1k
  • 1
  • 25
  • 39

Forcing the consistency of $ZF$ from a fragment of $ZF$

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

If $\mathfrak M$$\vDash$$T$ is consistent (where $\mathfrak M$ is the ground model) then for $\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.